a crossbow would work in space for the same reason a rocket works

it is newtons second law which states that for every action there is an equal and opposite reaction

hence the action of drawing back the string of the bow thus bending the bow bar stores potential energy in the elastic bar material. on releasing the string, releases the energy. which is imparted via the string to the bolt. the action of the string is a reaction of the bolt. in a rocket it is the potential energy of the chemical fuel when burned and ejected  which reacts against the rocket body

Yes in that the mass of the launching mechanism (the crossbow and suited up human) will react in the opposite direction but the bolt will still move away from its original position in the intended direction. But no in that without air to act on the feathers at the rear of the bolt it would tumble in flight because of unequal forces imparted by the crossbow, rendering it less likely to be facing in the right direction when hitting the target.