Yellowstone in early August?

tsetsaf's link above is good but it should be CG INSIDE Yellowstone NP. By putting in West Yellowstone or Wapiti you will find a bunch more that meets the OP's requirements.
We do three month out West without reservations staying mainly at USFS CGs that do not take reservations. Not a problem if you show up early Monday, not Friday afternoon.

Check out Baker Hole -- National Forest. About 5 miles north of West YS. Lots of room. A few Full Hookup. Most dry.

I have stopped there 3 times with out reservations.

GPS N44° 42.360' W111° 06.060'

Red Rock is fine but not likely to have open spaces. We were there in mid-September one year and it was packed. Early August is still in Yellowstone's prime time. I'm not into serendipity, so we have never tried to stay in/near Yellowstone without a reservation, but if you are able to arrive at the non-reservable CGs in the park by 9:30-10 AM, you may get one. For that time of year, though, I do recommend reservations, even at the CGs outside the park.

Last year we dry camped 4th of july week in Yellowstone and had our pick of sites and campgrounds. Literally none of the non-reservables were full. If all else fails you can always pop outside the park to a private or forest service campground. If you aren't aware already this is your Best resource for all cgs in and around Yellowstone

Side note... if all sites are full you can take a handicap site for one night but must share it should a handicap vehicle arrive later.

We decided not to stay in the Park and chose Red Rock RV Park only 20 min. from the West entrance. For us a great choice not only because of the CG and close proximity to Yellowstone but also we were on our way to the Oregon coast and on departure when getting back to the highway just took a right and being past Yellowstone and the traffic headed out.

Red Rock RVPark

Yellowstone

Safe travels.

it's really hard to get a campsite in yellowstone without a reservation, but we have, in the past, been able to get a couple days of no hookup with just a few days ahead call. but do call if you can.
or you can try your luck in west yellowstone which has several rv parks etc.

If you don't mind driving 30-40 miles, drop down to Colter Bay on Jackson Lake and "dry" camp at the campground. It's about 40 miles, more or less, from Colter Bay to the South entrance and about 20 miles past Lewis Canyon and Lake to get to the junction for Lake or Old Faithful. We do that quite a bit as we tow our FW and boat to Jackson Lake, park in Colter Bay Campground and fish. Then, if we are a little tired of fishing then we may go to Yellowstone for a day trip. It's not too bad, but folks on here will tell you to go to West Yellowstone because it's a shorter drive into the Park, and it is, but I don't think it's anymore convenient than Colter Bay. If Colter Bay is too far, might try for Grant Village, it's in the Park close to the junction I noted above and has "dry" camping on a first come/first served basis I believe. If you get there sometime mid-morning, you probably shouldn't have a problem getting a spot in August, same at Colter Bay.