The closest Forest Service camping can be found at La Wis Wis Campground ($18–20 tents, $36 RVs), seven miles northeast of town on Highway 12.
Also fairly close is the primitive and free Summit Creek Campground, 12 miles north of town, and just east of the Highway 12 and Highway 123 junction on Forest Road 2160. No trailers are allowed here.
Instead, park and hook up at Packwood RV Park (12985 Hwy. 12, 360/494-5145, $30). This wooded property is within walking distance of Packwood [1] and has on-site laundry and showers, plus cable TV.
Links:
[1] http://www.moon.com/destinations/washington/south-cascades/white-pass-scenic-byway/packwood