Up to the state, but federally, it's not a firearm. Legally, black-powder muzzle-loaders are not firearms, either, and a felon can lawfully own one under the fed's eyes.
Again, may vary by state.
Regulating crossbows is incredibly ignorant and stupid.
A felon can build an effective shotgun for $8 in an hour or two. The 12 gauge slamfire is more effective than most crossbows, and can be reloaded much faster.
http://gunwatch.blogspot.com/2016/08/mn-minneapolis-gun-buy-back-works-for_29.html
Slamfire shotgun shown at link.