Unless you’re looking for RV camping, it’s better to drive a ways up the Paradise Valley [1] to find a site [2], but in a pinch there are private campgrounds in Livingston [3].
About 18 miles south of town, Yellowstone’s Edge (3502 Hwy. 89 S., 406/333-4036, www.mtrv.com [4], May–Oct., $40 RV) is a particularly well-equipped and well-situated RV park.
The Forest Service (406/222-1892, www.fs.fed.us/r1/gallatin [5], $30–35) rents several cabins in the Livingston district. Not all cabins have electricity or drinking water; most have wood heat.
Links:
[1] http://www.moon.com/destinations/montana/the-missouri-headwaters-and-south-central-montana/livingston-and-the-paradise-valley/paradise-valley
[2] http://www.moon.com/destinations/montana/the-missouri-headwaters-and-south-central-montana/livingston-and-the-paradise-valley/paradis/camping
[3] http://www.moon.com/destinations/montana/the-missouri-headwaters-and-south-central-montana/livingston-and-the-paradise-valley/livingston
[4] http://www.mtrv.com
[5] http://www.fs.fed.us/r1/gallatin