Not sure what "basic amenities" means to you, but we always stay at the Gros Ventre NPS campground when visiting the Tetons. They have bathrooms with running water and flush toilets, but no showers unless it's something new. One loop with electric, other loops with no hookups. Don't know whether they sell ice, but they do sell firewood.
If you want to go cheaper you can try the Atherton Creek Forest Service campground on Slide Lake about 5 miles east of the little town of Kelly. No hookups and their bathrooms are likely pit toilets. The camp host sells firewood.
About 5 miles further out on the same road, which becomes gravel after Atherton Creek, is the Crystal Creek Forest Service campground. Definitely a one hole pit toilet there.