All you need is a normal water gun hooked up to a tube in your arm and you have an auto-filling blood gun with a few pints of ammo.

According to the professional vampires at the Red Cross, a normal person can safely donate a pint of blood every 8 weeks. However, that is just to have large safety margin, as the red blood cells are replaced in as little as four weeks:

How long will it take to replenish the pint of blood I donate?

The plasma from your donation is replaced within about 24 hours. Red cells need about four to six weeks for complete replacement. That’s why at least eight weeks are required between whole blood donations.

As with a normal blood donation, blood pressure alone will fill up the gun's reservoir in a few minutes:

Donation

We will cleanse an area on your arm and insert a brand–new, sterile needle for the blood draw. This feels like a quick pinch and is over in seconds.

You will have some time to relax while the bag is filling. (For a whole blood donation, it is about 8-10 minutes.

Since a pint of blood can be drawn in 8-10 minutes with blood pressure at a resting heart rate, we can assume the process will go much faster when heart rate is up during a fight. Even quicker if the gun includes a pump to manually pull the blood.

So, at a minimum you can safely have an auto-filling blood gun with a pint of ammo. However, you can definitely do more than that without dying:

Hypovolemia is the technical term for 'bad stuff from blood loss', and is something we want to avoid. According to this paper a person may enter hemorrhagic shock(described as 'rapidly fatal') after losing 40% of their blood in a short period of time, but anything less than 30% has relatively minor issues such as anxiety and slightly reduced oxygen delivery to muscles and organs. That paper also claims that the average person has 70 ml of blood per kg of body weight, so your hero has about 5600ml/11 pints of blood total. This means the hero can safely lose 1 liter/2 pints of blood without much issue, 1.5 liters with some lethargy and anxiety problems, or 2 liters with a serious risk of death.

In short, your gun can pull about 2 liters/4.25 pints of blood max before killing the user, so let's say your gun has 1 liter of ammo at a time to be safe. The first water gun I found online holds about 22 ounces/650 ml/1 pint of water and can shoot several blasts of water up to 34 feet/10 meters away, so you can definitely make a blood gun with a 1 liter reservoir and a range of several meters.

You'll need about a month to recover each pint/half liter of blood you use, so try not to get in too many fights.

Addendum on preventing clotting: Something I didn't consider until it was pointed out in a comment: these gross blood guns are going to clog from coagulating blood. The fresh stuff in the reservoir should be good for a bit, but the barrel will clog pretty quickly and the rest will soon follow.

To solve this problem all you need is a tank dripping anticoagulants into the blood reservoir, essentially turning it into a blood bag. There are many types of anticoagulants to choose from, so as long as you have a steady supply your gun should last longer than your body.

This is more of an attempt to state the obvious.

The most simple approach to this problem is not to use a syringe that actively draws blood and shoots it. This is horribly impractical, slow, inefficient, painful, unwieldly, and their is no control against drawing too much blood.

It makes a lot more sense to draw the blood out before hand, and keep it in a blood bag. Keep the blood active in the same way that blood banks do. Collect blood into a bag that has CPDA-1. The solution has Citrate (an anticoagulant), Phosphate, Dextrose, and Adenine. Have an attached heater to keep the blood around 40C so that it stays viable for up to 35 days.

https://www.ncbi.nlm.nih.gov/pubmed/7147334

Now you have addressed the blood part.

Next feed this blood into a high pressure water gun, or power washer. Now the ballistic part of this system is resolved.

The only thing you have left to worry about is running out of blood.

Sounds like blood bullets are essentially paintball bullets. So pretty much the exact same rules apply.

But let's get slightly more sci-fi. In theory, you could have a gun that extracts blood, freezes it instantly with liquid nitrogen, then fires it with some sort of gas propellant (like CO2).

The "bullet" would hit like a low speed musket round. Accuracy and range would be trouble, but it could definitely penetrate an unarmored opponent. A fancy enough gun could pump the blood into a mold before it's frozen into a bullet shape, then loaded and fired. You might even be able to use some rifling (but this generates heat, which would melt the ball)

This is a pretty worthless gun overall though unless your plan is to spread disease or something.

-Edit

As pointed out by the comments. You could use the nitrogen that froze the blood to also fire it, giving it far more power. Enough to actually function like a real bullet. Your accuracy would still be limited, but your range and hit power would be huge and most likely lethal.

For starter, I think your first concern is to avoid at any cost massive uptake of blood. Since during the peak of the fight it's easy to forget such details, I would go against "charging while fighting", and rather opt for a "charging before fighting".

Basically your guy would pump out his blood before the fight, maybe while still in his HQ, store it into a suitable vessel with anticoagulants, and then use it when needed. In this way your hero can also partly recover the removed blood before the fight, being fitter.

It's worth nothing that:

Class I Hemorrhage involves up to 15% of blood volume. There is typically no change in vital signs and fluid resuscitation is not usually necessary.

Class II Hemorrhage involves 15-30% of total blood volume. A patient is often tachycardic (rapid heart beat) with a reduction in the difference between the systolic and diastolic blood pressures. The body attempts to compensate with peripheral vasoconstriction. Skin may start to look pale and be cool to the touch. The patient may exhibit slight changes in behavior. Volume resuscitation with crystalloids (Saline solution or Lactated Ringer's solution) is all that is typically required. Blood transfusion is not usually required.

Class III Hemorrhage involves loss of 30-40% of circulating blood volume. The patient's blood pressure drops, the heart rate increases, peripheral hypoperfusion (shock) with diminished capillary refill occurs, and the mental status worsens. Fluid resuscitation with crystalloid and blood transfusion are usually necessary.

Class IV Hemorrhage involves loss of >40% of circulating blood volume. The limit of the body's compensation is reached and aggressive resuscitation is required to prevent death.

Since you want your hero to be active during the fight, I would stop at class I, meaning 15% of about 5 liters, that is 0.75 liters.

I would then use something like a paintball gun, to shot blood loaded projectiles which splash on impact, increasing the chances of hitting the skin.

With 0.75 liters of blood you can fill a fair amount of projectile, and maybe even more if you can slightly dilute the blood.

If you want to shot just liquid blood, than have a suction system connected to the vessel, and shot as long as you can use those 0.75 liters. But that would limit the maximum reach and also the pace of fire.

Don't use a squirt gun

I'm honestly surprised this hasn't been mentioned yet - squirting large amounts of blood mid-battle sounds like a horrible idea. The water gun suggestion in particular will only last for maybe a couple dozen shots - and tyical water guns have a comparatively slow shot speed which, along with the inaccuracy of the fluid jet due to turbulence, makes dodging relatively easy (think of fights using these guns during a hot summer - even in a relatively cramped garden, where combatants are mostly prioritising offense over defense and dodging, a non-negligible portion of shots will still miss). And then you have to recover for a month!

The "paintball" (bloodball?) idea is much more manageable. According to Wikipedia, paintballs are typically around 17cm in diameter, giving a volume of around 2.57 cubic centimeters, or about 390 paintballs in a single liter. Much better, as you have almost 400 shots per month, but you're still going to be feeling slightly weak most of the time.

Since you mentioned that "any amount" of blood will do, I think we can do a lot better.

Aerosol

I wasn't able to find good data online, but if you could install a mechanism to properly spray a very fine mist of blood, you could cover a significant area around you. It would have shorter range than a directed shot, but would be amazing in close quarters against multiple enemies - you'd only use somewhere on the order of 1ml of blood to reliably hit everything in your immediate vicinity.

This could also be useful for defense: spray a fine mist all around you, or - even better - at a choke point, and until the suspension settles (which could take a couple of minutes if the mist is fine enough, or even longer) this becomes an impassable area that kills any enemy that tries to go through.

The disadvantage is that this would be of limited effectiveness in windy conditions. Although slight air currents, if favourable, could actually help spread the mist - rendering it less useful for setting up defensive blocks, but more effective at spreading to kill everything around you. I don't know if this would actually be feasible in practice, but spraying some aerosol into an indoor ventilation system might perhaps be a great way to mostly clear out a building.

Aerosol bullets

Spraying blood mist is very limited in range but covers a wide area. Bloodballs/pellets have excellent range but need careful aiming, and a miss means wasted blood: you can't really sustain any sort of automatic firing. (Squirt guns, again, have neither and should really not be used.) How about a capsule filled with pressurized gas and blood, that upon impact will shatter like a paintball and cover everything in a wide area around it in a fine layer of blood? Or even exploding bullets - it's hard to find real-life examples of these since various treaties ban their use on low-caliber weapons, so these mechanisms are usually found on anti-vehicle armour-piercing ammunition, but the fact that they were deemed worthy of a ban as early as 1868 makes me think they're very feasible to construct. Add a payload of blood, and you only need to hit some hard surface (such as a wall or floor) near the enemies to spray them.

This makes shooting require significantly less accuracy - meaning less wasted blood. Moreover, you can hit several enemies in one shot, and - if the resulting spray is fine enough to function as a proper aerosol - this can be used to set up killzones mentioned above at range, and render an area deadly (restricting the movement of your enemies) even on a complete miss.

Poisoned bullet

Instead of making a thick bullet with a complicated explosive mechanism, you could simply fire blood-delivering bullets. Generally that means something like a hollow bullet filled with blood.

While less effective than aerosol-spraying ammunition (since you lose the benefits of AoE), this sounds much simpler, and while I don't know much about bullet ballistics I feel like this would make it more feasible to, say, deliver blood through a sniper rifle - or even make an armour piercing blood bullet. It also allows you to very carefully control the amount of blood delivered - you could place a tiny droplet into a tiny dimple on the bullet, and you'd be sure it would touch its target.

The only issue I can see is with clothing absorbing the blood before the bullet impacts the wearer, if your enemies wear that.

Conclusion

The main things I focused on was reducing the amount of blood needed per shot, and increasing the efficacy of each shot. Since there's no mininum amount of blood needed for effectiveness, any blood not strictly meant to come into contact with a living enemy is wasted. Using these methods will probably use little enough blood that you could engage in intensive combat regularly with almost no negative side-effects.

Mind you, these solutions are technologically advanced - you'll need a proper aerosol mechanism that creates as fine a mist as possible over as long a range as you can achieve (a non-trivial engineering task), and you'll probably need your own very specialised ammunition for the explosive bullets that would allow you to fill them on demand - or a hyper-specialised gun to coat bullets with blood in a precise way that uses the minimum amount of blood while still ensuring it's reliably delivered. This isn't something you could jury rig together in your backyard from a needle and a hose pipe. This also departs significantly from your initial image of "squirt some blood". So these methods may not suit your tastes - but they're all technologically feasible given adequate resources and achieve what you asked in your question in a very efficient way.

A note on connecting to the blood supply

A simple way would be to install a blood plug with a fail-shut valve. The gun would connect to the plug, which would open the valve. Simply make the coupling easy to disconnect, and any impact or injury will dislodge the connection (shutting the valve) before damaging any of the gun's parts. Sure, this might result in accidental disconnects in the middle of heated combat, but that's better than bleeding out. The connector could even be magnetic to make it extremely fast and easy to simply pop back into place when it gets dislodged.