Yes, and no.
Yes, in the sense that by paying corporate taxes in their state, the teams are funding the general pool of workers compensation claims, however the state manages it.
Also generally speaking, yes, pro athletes are eligible to receive worker's comp from injuries resulting in play, and their players do make successfully compensated claims.
No, not every claim is successful. Due to the variance in state laws, some players may not be able to claim for anything but a specific career ending injury. This is not something that's unique to sportsmen and sportswomen, though.