The closest public campgrounds are in Riverside State Park [1] (509/456-3964, www.parks.wa.gov [2]) six miles northwest of town. Tent sites ($14; no RV hookups) are available year-round. Mt. Spokane State Park [3] (509/456-4169, www.parks.wa.gov [2], $14), 30 miles northeast of Spokane [4], has tent sites and is open May–October.
Private RV parks include Alpine Motel RV Park (18815 E. Cataldo, 509/928-2700), Alderwood RV Resort (14007 N. Newport Hwy., 509/467-5320 or 888/847-0500), and Yogi Bear’s Camp Resort (7520 Thomas Mallen Rd., 509/747-9415 or 800/494-7275).
Links:
[1] http://www.moon.com/destinations/washington/spokane/sights/riverside-state-park
[2] http://www.parks.wa.gov
[3] http://www.moon.com/destinations/washington/spokane/sights/mt-spokane-state-park
[4] http://www.moon.com/destinations/washington/spokane