Would a site where people pay a monthly subscription fee to participate in a random prize giveaway be illegal in Ohio?

I don't know Ohio law specifically, but I would very much doubt that such an unauthorized gambling site would be legal.  This is a highly regulated area.  I used to represent a company which ran internet lottery games and, while the regulations differed somewhat from state to state, they did not charge any fee for entry into the game.  They made their money from collating the information provided by subscribers, and marketing the resultant lists.