Check out this Yellowstone site from the park itself. It gives all the campground information along with 'clicky's' for things to do, etc.
Yellowstone Nat'l ParkWe've been there often and even with our 40' motorhome we stay IN the park for the best experience. We enjoy the rustic campgrounds such as Madison, Mammoth and Bridge Bay. Some are no reservations and for those you need to stay nearby the previous night and get to the campground early morning when people leave. Don't try to pull in in the afternoon and expect a non-reservation site. Some accept reservations.
During your visit be sure to plan at least three nights in the adjoining national park - Grand Teton. It's beauty is altogether different from Yellowstone's. The hiking trails are awesome and of all abilities and distances. Take a gentle guided float ride down the Snake River and you might even see a moose in the water with you. It has a full hookup RV park called Colter Bay RV and also some very nice dry campgrounds - Colter Bay Campground is right next to Colter Bay RV and the campground has 300 wooded sites. Further south in the park and closer to the town of Jackson is Gros Ventre campground - also 300 scenic sites. Both have dump stations and water fill.
For the kids, a day or two checking out nearby Jackson is fun. There's also a rodeo in town. Major grocery stores and cheaper fuel is also in Jackson.
It sounds like you're still working and have limited vacation time but no matter how much time you have for this trip, even a few days in each place is much better than none at all. The kids will have great memories. You can always return!
Have a great trip.