well All I can suggest is you call the Ohio DCNR and ask, and get A REAL answer from the state, anything we say here can or might be incorrect

here in PA where I am from we have a special draw/season for them
BUT any animal that is in my state that the state doesn't consider here, like say a cougar
if you were to shoot it you would be fined up the wazzoo! as they could consider that an illegal kill
here in PA and many other states i know of, any animal that doesn't have an actual hunting season is a protected critter and thus would be an illegal killing, so I highly suggest you ask the officials and not a forum member
not saying anyone would steer you bad, but better sfae than sorry

and last, Bobcats are cool critters, I highly enjoy seeing them in the wild way cool cats in my eye's