Simply put, it depends on the kind of bullet. To propel a round of ammunition out of a gun, an explosive activated by the trigger (connected to the hammer) of a firearm ignites a small amount of gunpowder in the bullet casing to create a mixture of expanding gases that accelerate the bullet out of the barrel. The key is in how the explosive ignites the gunpowder upon impact from the hammer.

Particular recipes for explosives are naturally protected, but suffice it to say that some explosives rely on a spark created by static electricity accumulated by motion of the hammer, and the spark can only ignite the gunpowder if oxygen is present. Note that oxygen gas is not typically sealed into bullet casings because they could potentially ignite too easily, for example if the round of ammunition were dropped.

In space, a static electric spark would find no oxygen gas to ignite the gunpowder and the bullet would not fire. The spark itself is also likely to be more short-lived in the absence of oxygen, though it could exist briefly since metal particles can be heated to glowing temperatures by friction. That said, modern ammunition contains explosive igniters with oxidizers. When impacted by the hammer, these materials not only reach a high temperature, but they also generate oxygen. This kind of bullet would be able to fire in space because once oxygen and heat are present, the gunpowder can ignite and produce the propellant gases that push the bullet out of the barrel.

Remember, ammunition rounds contain volatile, flammable substances, the mishandling of which can have disastrous consequences. Please don't investigate this problem at home.