Not that nearby, but there is a campground at Croton Point Park. It is a huge park located on the shores of the Hudson River. About a 30 minute drive from Kitchawan
You can camp at Ward Pound Ridge Reservation, which is a state park. Just have to set it up before going.
There's no camping close enough to be able to walk in or easily drive through.
I think there is camping on beat mountain and Putnam valley
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