Depends on the state, Georgia Code § 16-11-131(a)(2) defines firearm as "any handgun, rifle, shotgun, or other weapon which will or can be converted to expel a projectile by the action of an explosive or electrical charge." I believe that means archery equipment is not inlcuded. But I don't know if it's legal to hunt with a crossbow in Georgia, only that you shuoldn't have any trouble owning one.
In most states it is legal for felons to bow hunt. Cross bows however usually require you to be handicapt and have a dr's note on your person. Just google your state laws concerning such an issue. Or call the DNR