The American Alpine Club operates the Climbers Ranch in Grand Teton National Park. Non-private bunkhouses, communal cooking (optional), hot showers. $12 per person per night for non-AAC members. The Climbers Ranch is ideally situated for GTNP, but is not easy walking distance from the town of Jackson.
http://www.americanalpineclub.org/pages/page/90
The Hatchet campground on Bridger-Teton National Forest is situated at the foot of one of the most spectacular dirt roads I've ever run on. Again, not walking distance from Jackson. If one of your party has a YMCA "away" membership, you could probably make a trip into Jackson for showers every second or third day. All food in campgrounds on the B-TNF has to be in bearproof storage.
http://www.fs.fed.us/btnf/teton/camping/buffalo_cg_details.shtml#hatchet_campground
The earlier posters are correct, what you're looking for is not readily available and affordable around Jackson. You might check out the towns of Alpine, Wyoming, West Yellowstone, Montana, or Dillon, Montana instead. Even then a private KOA-type of campground might be your best option.