Kind of a grey area last I read the ORC and OAC that contains the hunting regs.
For deer gun season, you are allowed a shotgun, handgun, or rifle in a strait wall caliber, I'm sure you know, but those are how it's described in the section of the OAC that pertains to the Dept. of Wildlife.
The rub is that under Ohio Revised Code, SBSs and SBRs are defined as "Dangerous Ordinance", not legally the same as "Rifles" or "Shotguns" in our legal code. So an argument could be made that dangerous ordinance is not permitted for taking deer, but there really isn't anything definitive I'm aware of that sets it apart since its a discrepancy between the definition in the Revised Code (codified laws) and the Administrative Code (rules written by the ODNR in this case).
For what it's worth, Silencers are also considered dangerous ordinance here, and it took an addition to the ORC to specifically call them out as allowed for use while hunting.