There are no public campgrounds in the immediate vicinity of Seattle, but Fay-Bainbridge State Park has year-round campsites on Bainbridge Island, just a short ferry ride away. You'll need a car or bike to get to the park from the ferry dock on Bainbridge, six miles away.
Other state park campgrounds around Seattle are in Federal Way (Dash Point State Park), Enumclaw (Kanaskat-Palmer Recreation Area), and Des Moines (Saltwater State Park). The closest Forest Service campgrounds are in the Cascades, at least 35 miles east on I-90.
As for private RV parks, the best bets for centrally located urban parks are the Issaquah, Lynnwood, and Renton areas.
© Ericka Chickowski from Moon Washington, 8th edition