Yellowstone RV Park (406/848-7496) and Rocky Mountain (406/848-7251) are RV campgrounds right in Gardiner [1].
Eagle Creek is the nearest Forest Service campground; it’s four miles northeast of town on the road to Jardine.
Links:
[1] http://www.moon.com/destinations/montana/the-missouri-headwaters-and-south-central-montana/livingston-and-the-paradise-valley/gardiner