It's very easy to get into Colter Bay campground. It usually doesn't fill up until later in the day, if at all. Most of the sites will allow generator during proper hours but don't have water or sewer. One loop does now have electricity on very large spots, but mostly for handicapped campers, although if not taken later in the day, they will rent to regular people. Colter Bay is about 45-50 minutes from the South entrance to Yellowstone and a person can see Yellowstone from Colter Bay, it just takes a bit of driving. We usually get up pretty early and sightsee in Yellowstone for the day and then get back to Colter Bay pretty late in the evening for a nice meal in one of the prettiest spots we have ever seen. Unless you are an avid hiker and just have to hike all over the place, a person can take two to three days and see most of Yellowstone that can been seen from a drive. The lower loop takes you to Yellowstone Lake, Canyon, over to Norris, down to Old Faithful and then back south to Grand Teton. The upper loop goes over Dunraven, over around Lamar, up to Mammoth and then back down to Norris, Old Faithful and back to Grand Teton. Yes, there is a lot of mileage and a lot of animal jams, but why get in a hurry, you are in a part of the world that is wondrous and spell-binding. Just take your time and enjoy. As for Grand Teton and Colter Bay, it is a beautiful place and you can take a cruise on Jackson Lake on the marina cruise boats, drive around the Park and see Jenny Lake which is beautiful, hike to Hidden Falls on the other side of Jenny, go over to Moose Junction to the Craig Thomas Visitor Center, drive over to Gros Ventre to see the results of one of the largest landslides in history, and look everywhere for wildlife, it's there if you are lucky. Have a great trip!!!!!! By the way, if you have the Senior Pass for national parks, the campground cost, for the non-electric sites, will run you about $12 per day, what a bargain.
Try Colter Bay in the Tetons which is a short drive from the south entrance of YS. They have vacancies every day and usually don't fill up until later in the afternoon-even in July. No hookups but they have a water fill and dump. We actually enjoyed this more than staying inside YS (except for Madison, but that is on the west side).
At this point, you may be out of luck for anything requiring reservations. I would look in the Teton/Jackson area for possibles. If you don't need to be at the Park, try something on the "Front" side of the Tetons like Victor, or Swan Valley, or my preferred area Island Park.
If you want further south you'll be in Grand Teton Nat'l Park and Colter Bay RV for hookups or Colter Bay campground - no hookups but dump station and water fill. The campground has 300 sites, rarely fills, and accepts no reservations.
Grant is the closest inside the park to the South Entrance. However, we stay in Bridge Bay and have "easy" access to everything on the southern route.
If you main visit area will be geysers, suggest you try Madison.
Or, go just outside the park to Flagg Ranch. It is situated between Yellowstone and Tetons.
May be a bit more helpful if you explain what you are wanting to see in the area. Yellowstone is HUGE. Not really possible to stay at the south entrance yet be able to spend any amount of time on the north loop.