Chincoteague National Wildlife Refuge is the essential stop, protecting the Virginia end of Assateague Island. It combines beach access, marsh drives, walking trails, lighthouse views, and a good chance of seeing the Chincoteague ponies at a respectful distance.
Treat it as a wildlife refuge first and a beach second. Bring binoculars if you have them, watch the tide and mosquito reports, and expect access rules to change when birds are nesting or storms have shifted the sand.