You can get an EZ-Pass transponder from any state in the EZ-Pass consortium, including Ohio. They all are the same system. Some (maybe all) member states have walk-in centers if you don't want to take the time to get one via mail or whatever. Different states have somewhat different fee structures for the EZ-Pass account, whatever they call it, and sometimes discounts available for their state.
If you'll be traveling the New York Thruway much with a motorhome, they have a special motorhome plan that is worth considering and setting up the account with New York. Basically, with the plan, you pay the equivalent auto rate for a motorhome, which works out to about half the cost on the thruway. If you set it up with a credit card and email statements they have no annual fees.