There is a campground East of Seattle, on I-90. IIRC, it is fully paved, and has pet restrictions.
There are a couple of campgrounds near Poulsbo, on the Kitsap Peninsula. You will have to take the Bainbridge Island Ferry to get to Seattle.
It has been close to 15 years since we stayed out there, so things may have changed.
The current Woodalls Directory would be your best bet to find a place. However, if you look at "Seattle" you may not find much. It would be best to check the map for the towns around Seattle, and check each one individually.
Good luck.