Camping around Newport RI.
There is not much around this area. I will list three in order of my preference.
Meadowlark, Middletown, RI 401-846-9455, about 15 minutes to Newport.
Melville Ponds Campground, Portsmouth, RI about 20 minutes to Newport.
Paradise Motel and RV park, 401-847-1500 also in Middletown, RI
You might want to get reservations soon, they book up very quickly. They are all small sized campgrounds, top two are nothing special but are very close to Newport. They third one is very small and I would only stay there if nothing else was available.
Lots of very nice campgrounds in Western RI but all are about 35 to 45 minutes from Newport via the Newport/Jamestown bridge.