Eric - good advise from previous member posts, and I'll add a few of our favorites (not in any real order):
They do allow beach driving in select spots up and down the East Coast, but very few that actually allow 'camping' on the beach. There is a way to stay on the beach, but you have to work around the restriction: EX: Island Beach State Park in NJ (near Seaside Heights) you're allowed to drive the beach for surf fishing. You're allowed to be on the beach all day/night if you are actively fishing. So you'll need to bring two poles and leave the line in the water. Last I heard you are still allowed to have fires. Recommend you check the 'restriction' at any location to see if this are still in effect.
Seawall Campground in Acadia National Park (Mount Desert Island) is quite nice, especially since you hear the waves crashing on the rocks through the night .
Lamoine State Park is about 20-25min to Acadia, but you can have Lobster delivered to your camp site. It's on a Bay, and when low tide hits your entertainment is all the people that walk out on the exposed muddy bottom to clam. Really a good time watching them struggle to pull their feet out for the next step.
If you head through from Assateague Island up through Maryland Eastern shore to Rehoboth Beach you can take the Lewes Ferry to Cape May (cool little Town). From there you can head north towards Atlantic City.
National park campground on Ocracoke Island
OUR FAVORITE: Head to Hatteras Island if bad weather is immanent. We have a blast at Frisco Woods campground - they have RV sites are right on the edge of the beach. It's our favorite spot. If you are lucky, there will be a squall - that's when all the locals (and many storm chasers from all up and down the Eastern Seaboard) carrying sailboards & windsurfers plunder the place...it's like watching a huge group of kamikaze daredevils. You're also able to make a lot of new friends if you bring extra adult beverages to share at the camp fire, The best camp sites are 51, 52, 53 and c21 thru 25 since that's where everyone deploys.
Cape Lookout National Seashore has primitive camping beachfront, and will need 4-wheel drive to get around.
Camping is available only in the Maryland side of Assateague Island, and you can purchase a permit (good for a year) to take your 4x4 van on the beach to the 'bullpen'...only fully self-contained 4x4's are allowed to camp there. Only problem is the soft sand. I've tried twice and wasn't able to make it because of the van weight (sank to the axels both times, but did manages to maintain composure by getting myself out). The main Campground is fun with the wild horses which leisurely walk right into your camp (they roam free - just secure your food!).
Since you'll be close (somewhere in your 'plans') I'd encourage you to organize a slight detour to Mount Washington NH and ride the Cog Railway to the top. It's the 2nd steepest railway in the world & truly an experience! Believe the Weather Observation station at the top also holds several temperature records in the US for the highest wind (if I recall it was over 230mph), and coldest wind chill (over -100 degrees). Many fun hikes at the top with outstanding scenery. There is also a very cool road that takes you to the top, but they will not allow your Van to make this trip.
Either on you way up, or your way back you try driving Skyline Drive in Virginia. Great scenic turn off's all over - hiking trails everywhere. Several nice Campground along the road as well.
You may want to PM ctb since he's the beach bum on this Coast, and probably not divulging all his 'favorite' spots to protect them. But I think he would agree, OBX has some of the nicest beaches. I personally prefer the Acadia area up in Maine...I just prefer the cliff's, crashing waves on the rocks, etc.
If you're close to Annapolis in your travels you've an open invitation to our driveway, with water taxi 1/2 block away to take you into Historic Annapolis - Great Seafood & Bars! Tour the Naval Academy, and you'll only be 20-25 minutes from DC if you're interested in visiting some of the sites there.