Ohio allows them, all season long and they are legal for any game other the waterfowl. there are restrictions on them but its not anything major, most crossbows full within the regs.
Heres the Ohio Regs I could find in a hurry.
Unless otherwise indicated, game may be taken with longbow, crossbow, or any caliber handgun, rifle, shotgun (10 gauge or smaller), or airgun. Be safe and choose the appropriate method for the species.
Longbows and crossbows may be used to take legal game. However, crossbows may not be used to hunt migratory game birds. Longbow hunters may use a hand-held mechanical release. Crossbows may be cocked with a device, but must have a working safety and a stock more than 25 inches long.
Arrows for longbow and crossbow must be tipped with a broadhead not less than 3/4 inch wide while hunting deer or turkey. The arrow tip must have a minimum of two cutting edges which may be exposed or unexposed.
It is unlawful to hunt deer or turkey with a longbow having a draw weight of less than 40 pounds, or with a crossbow having a draw weight of less than 75 pounds or more than 200 pounds.
Poisoned or explosive arrows are unlawful. While hunting, it is unlawful to have attached to a longbow or crossbow any mechanical, electrical, or electronic device capable of projecting a beam of light.
taken from this site
http://www.ohiodnr.com/wildlife/dow/regulations/hunting_general.aspx