Thank you for your question. Here is a page listing all of the camping sites on the Outer Banks. I would say that the closest one to Nags Head would be the one just south at Oregon Inlet. It is run by the National Park Service, so it is pretty basic. I do believe they permit dogs.
Here is a page that details the Oregon Campground from the National Park Service. There is also a camping facility in Kill Devil Hills, but it is not on the ocean like the one in Oregon Inlet is.