Amazing Beach Camping (but a little pricey)
There is only one campground in then Assateague park and it's right on the beach. Get ready to get sand everywhere. Be sure to check the forecast before you go because if it gets windy you'll be sure to feel it. I visited in later October and there were very few campers but I imagine the campground fills up in warmer months. The park is small to there are just a few one-way roads leading in and out.
The campground was two loops with drive-in sites as well as walk-in sites that are more removed from the road. There are also cold water showers to help with some of the aforementioned sand.
You'll surely see at least a few ponies on your visit as well as other great wildlife. If you have the time I highly recommend taking the drive over to the Chincoteague side (about an hour) and exploring there. I rented a bike from the Bike Depot and rode it into Chincoteague for a few hours.