The closest public camping can be found at Millersylvania State Park [1] (12 miles south of Olympia off I-5 at exit 95, 360/753-1519, www.parks.wa.gov [2]). Tent ($14) and RV ($20) sites are available year-round. Make reservations ($7 extra) at 888/226-7688 or online.
Approximately 20 miles southwest of Olympia, Capitol Forest [3] (www.wa.gov/dnr [4]) has an abundance of primitive campsites (free; no water). Contact the Deptartment of Natural Resources at 360/902-1234 or 800/527-3305 for details and a map.
Olympia [5] has an abundance of private tent and RV camping facilities, including several just a few miles from the center of town. Best among them is Olympia Campground (1441 83rd Ave. SW, 360/943-8778, www.olympiacampground.com [6]), set on 10 acres of lightly wooded grounds. There are a ton of amenities, including a convenience store, a gas station, an ATM, a laundry room, a heated outdoor pool, and a game room with free Wi-Fi.
Links:
[1] http://www.moon.com/destinations/washington/south-puget-sound/olympia/sights/millersylvania-state-park
[2] http://www.parks.wa.gov
[3] http://www.moon.com/destinations/washington/south-puget-sound/olympia/recreation/capitol-forest
[4] http://www.wa.gov/dnr
[5] http://www.moon.com/destinations/washington/south-puget-sound/olympia
[6] http://www.olympiacampground.com