There are many RV sites for rent in Florida owned by individuals. I've owned several over the years. Not sure why you are limiting yourself to those type of sites but whatever. Around Naples, there is Silver Lakes Golf and RV Resort, Pelican Lakes (Class A), Naples MotorCoach RV Resort, Chrystal Lake RV, Everglades Isle Motorcoach (Everglades City), Outdoor Resorts (Chokoloskee). Around the Stuart area, 3 come to mind immediately - all on Hutchinson Island. They are Nettles Island, Venture III and Holiday Out. I believe there is a fourth - Windmill something or other just south of Holiday Out. Further north is The Great Outdoors near Titusville. There must be several hundred other parks in FL which have gone condo with individual lot ownership. The parks listed above and most, I'd guess, have rental offices and realty offices associated with them. Lot owners who wish to rent often "list" their lots with the rental office which then rents the lot by the day, week, month or season. These rents generally are "market rents". If you go to each park (bulletin board) or know someone in a park, you can usually find an owner that will deal direct splitting the rental commission cost with the renter so both parties net out better.
Finally, if I google "RV sites for rent Florida" I see a number of websites which purport to have listing of lots for rent by owner. Your signature says "Florida Treasure Coast". Have you checked some of the various parks when you've been down here? That, IMO, is the best way.