Dad's from NC, around Morehead City (which is near Cape Lookout). Mom and Dad had a beach house on the Outer Banks for over 20 years (sold it a couple of years ago) and we'd always bring our dogs and at least one cat.
Lots of folks bring their dogs to the beach - there is a leash law and Mom will literally chase after the thoughtless people who don't clean up after their dogs. The idiots don't seem to realize they could close the beach to all dogs if enough people complain about dog poop (and have every right to do so.)
Do a search on the web for vacation rental companies along the Outer Banks (Sun & Surf and Prudential are a couple of companies on the island where we rent - the whole island is called Bogue Island, with various communities like Emerald Isle, Indian Beach, Salter Path). Companies will list rental properties that allow dogs (usually a max of 2 and you have to pay a pet deposit). Off season is a LOT cheaper and if you don't mind walking a block or two, houses that aren't on the beach are cheaper than beachfront property but don't have as nice a view. It's a great place to visit!
