We camp quite a bit on the cape.
There are a ton of campgrounds on the outer cape near the national seashore but is only one campground that is walk to beach in the region of the National Seashore; North of Highlands Campground in Truro. It is nice and about a 15 minute walk to Head of the Meadow Beach in the National Seashore but the beach is not the best. They have a playground and rec room for the kids but neither are fancy, just simple and functional.
Our really close to the National Seashore preferred campground however is in P-town, and it's called Dune's Edge. They are mostly tent sites. There are very few amenities in this campground beyond the showers but the proximity to the National Seashore beaches and Provincetown is great and the sites are clean and reasonably well maintained.
There are a couple of other campgrounds near Marconi Beach and Nauset Beach but they primarily cater to the camper crowd and since we strictly tent camp we have never stayed at them. Maurice's is supposed to be nice.
Be aware that on the outer cape - P-town, Truro, Wellfleet, Eastham there are no campfires allowed in any of the campgrounds - no open flame of any kind because of the fire risk. So you must have a camp stove to cook at your site. You can have a fire on the town beaches and possibly on the national seashore but you have to get a permit the morning you plan to have your fire. Last time we got one the permit in Truro cost $5. You must bring a shovel with you to bury your coals after the fire is out.
If you are looking to have a campfire and still be in close proximity to national seashore beaches then Nickerson State Park in Brewster may be a good option. My kids just went camping there this past weekend. The campground is very clean and the mom on the trip said that the bathrooms were clean. They have a camp store and a pond. The also have direct access to the Cape Cod Rail Trail Bike Path which goes from Dennis to Wellfleet and passes thru the National Seashore. ETA - my kids were staying at Nickerson for a Boy Scout Bike trip. From camping area 7 (the farthest back in the park) it is 3.8 miles by bike to the CCRT and then just over 10 miles to Marconi Beach on the CCRT and then 2.9 mi to the actual beach.