Yes, gun cartridges will fire in space as they don’t require oxygen from the atmosphere (like a car engine).
The bullet is accelerated from the expanding gases out of the barrel so it would move even faster than normal as there is no air pressure working against the expanding gases.
The bullet wouldn’t experience negative acceleration from friction (air resistance) either so it wouldn’t slow down.
The gun itself would have to be designed to higher pressure of the expanding gases as there is no atmospheric pressure counteracting it.
Also note that the gun would be accelerated backwards by the expanding gases as well, it wouldn’t speed like the bullet as it’s more massive but it would be accelerated in the opposite direction.
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