It's probably not what you really want, but there is camping (or, perhaps more accurately, overnight RV parking next to a picnic table) at
Floyd Bennett Field in Brooklyn. There are no hookups or dump station Just getting there with an RV would be something of an adventure in itself, I suspect; the most obvious route via the Belt Parkway is not an option due to typical parkway vehicle restrictions.
It looks like it's about a fifteen minute bus ride to the subway, or an hour or so walk. Biking also appears to be reasonably practical at first glance.