You can also stay in Chincoteague and access the south end of Assateague Island through the National Park. No pets allowed in the National Park, though.
We stayed at Pine Grove campground on Chincoteague this past fall, full hookups, shady sites, a short drive from lots of cool stuff. About 45 min drive to get to the north end of the island from Chincoteague. There is a KOA and amusement park near the access to the north end of the island.
We left our pup in the camper one day, took him to the state park up north the next day, which is how we found out it was a 45 min drive to get to the north end, didn't do our homework on the National Park pet policy.
Tom's Cove campground on Chincoteague Island has waterfront sites, on the inlet, but reservation system is a little old school, have to mail in a check.
Tough to get an actual spot on Assateague Island, spots go very quickly when they come online.