The closest to Seattle is probably this one in Issaquah, right off I90.
http://www.ivrvpark.com/
Without commute traffic, 20 minutes to Seattle. Also, 20 minutes in the other direction to Snoqualmie Falls, a favorite tourist spot. I don't know about pets, it's too close for us to stay there, since we live just up the hill in Sammamish. It's not all that scenic, but my parents have stayed there in their 5th wheel and enjoyed it. You would take I5 to 405 to 90 east, but arrive on a weekend or mid day to avoid congestion.
An alternative to the north of Seattle is Lake Pleasant in Bothel, but I'm not familiar with it. You would pass it going down 405, and it looks nice.
http://www.goodsamcamping.com.s3.ama...ePleasant.html
In Seattle the usual favorites are Pike Place Market and the Space Needle, but a walk along the waterfront is fun, with a great aquarium, Ferris wheel, and other attractions. I would recommend driving to the Issaquah Transit Center and take the 554 bus to Westlake, then walk. That's how I get to work. The traffic in Seattle is nasty, and parking very expensive.