Tough to find
I have not camped there in years, but have several friends who do. Based on conversations with them, very few private campgrounds, several at state and county parks but difficult to get into. If memory serves here are a few...Hither Hills state park, Montauk; Smith Point county Park, Cedar Point county park, Campgound in Greenport which I think is private, Wildwood state park, Southhaven county park. Those are the ones I recall, but know little about how to get in, I remember some of them were on a "lottery" system with reservations needing to be made a year in advance. There may be one or two private campgrounds left on the island but not sure. Also, not sure what you are camping in, but several of these are limited in equipment type and size, and most have limited, if any, hookups.
Tom and Patty
The "Rode Crew"
2004 National RV, Seabreeze 1341, F-53 22K, Chevy Malibu toad.