Are there any lots available for rent on a monthly basis in this area?
Google "Buckeye Lake Tourism". This will give you several website sources including the Licking Co. V&TB. Any of these sources should be able to offer guidance.
Keep in mind that Buckeye Lake is still in the throes of a major overhaul of the dam which was deemed unsafe and is being rebuilt. You may want to visit the place and take a look before signing a lease because access to the lake may be questionable in its current state.
Great advice from B & G to Buckeye Lake