I'd suggest staying in one of the non-reservation campgrounds in Grand Teton Nat'l Park - Colter Bay is closest to Yellowstone and do a day trip into Yellowstone. I'd not normally recommend day tripping into Yellowstone because it's more driving in this large park but your main concern is getting a campsite. Colter Bay and Gros Ventre campgrounds in Grand Teton each have 300 non-reservable sites and we've always gotten a site with our 40' motorhome. No hookups but you can surely survive that for a couple days. They have water fill and dump stations. As a backup keep 'calling' (not doing it online) the Yellowstone reservable campgrounds as you approach the area and there might be a chance for a cancellation that you can snag.
If day tripping pack your cooler so you don't waste time in the park restaurants. Plan to leave the campsite early.... by 7am and you'll see more animals that way and will have a lot less traffic, especially for Old Faithful. Check for Old Faithful's eruption times and head for it first, if convenient.
