As far as I can tell wild turkeys are considered a game bird in Ohio and as such do require a permit to own and raise them. The fee for a non-commercial permit is $25 per year.
As a general rule if you live in an area that allows hunting of a game bird and requires a license to hunt it, you also are required to have a permit to raise them or have them in captivity. Some states like Indiana do not allow possession of Wild Turkeys at all.