There are two Delaware State Parks near (but not right on) the ocean beach - Cape Henlopen State Park north of Rehoboth Beach, and Indian River State Park south of Rehoboth Beach. Both are nice, and within walking distance to the beach, but do not have beach views, etc. Pets are allowed in the campgrounds, but I am not sure about on the beaches.
As others have noticed, there are two campgrounds at Assateaque - the Federal and the State. The State is closer to the beach, has some hookups, but does not allow pets. The National Park has no hookups, but allows pets. As mentioned above, driving on the beach is the best way to access it, but requires a permit.
Weather in Delaware/Maryland should be pretty decent through September, but is probably more reliable beach weather in Hatteras, which is another 6 hour drive south. The most popular campgounds seem to be Camp Hatteras and the KOA in Waves, NC which are both within walking distance over the dunes to the ocean beach - but no beach view. We like Frisco Woods Campground, further down Hatteras Island in Frisco, and it has water views, but on the "sound side", not the ocean. There are limited ocean views from the dune side sites at the National Seashore Campground in Frisco, but no hookups. Like Assateague, beach access is much easier by 4x4 vehicle, and a permit ($50) is required.