There are a number of campgrounds in that area, but if you're looking for someplace free there's always Camping World in Bowling Green. Probably a 30-40 min. drive to the cave, though.
If you're a member, Passport America has Singing Hills Campground in Cave City. PA members pay 1/2 price, although there may be restrictions. Don't know a thing about it except that it popped up on our DeLorme Streets and Trips map overlay.