The other thread is gone now, but I'm curious about this now.
It had never occured to me to think about taking any animal other than fish with bowfishing gear. In fact, I've never hunted beaver at all, at least not for the purposes of this discussion anyway.
Taking a peek in the IL regs, it appears that beaver is only mentioned in the trapping section, and there are definite season dates for them. Can they ever be hunted with a bow in IL, or are they only available for trapping?