It's going to be a bit further than your 100 mile round trip criteria but you have the option of getting a bike pass and taking the LIRR part way if need be.

Check out Hither Hills State Park out by Montauk. They are pretty booked from what I can see up through the first week or two of September (at least for longer stays) but you may be able to get a campsite for a night or two as people cancel all the time. Anyway, it's a mix of tents and campers/RV's and they have a bathroom w/showers. You are right on the beach - it's a pretty nice place with the sound of the ocean and the easy access to amenities in Montauk proper. If you do a search for Hither Hills you'll find a link to the online reservation system.