I've lived here 60 years now. There are four campground facilities in the Napa Valley. Skyline park and Calistoga fairgrounds have been mentioned. There is also an RV campground at the Napa Town and County Fairgrounds in downtown Napa and Boothe State park just north of St. Helena on highway 29. The fairgrounds has been redone and is a nice facility with hookups though you will not consider the view particularly inviting. Boothe is nice but generally booked in advance and has traffic noise from the highway, but a nice surrounding area.
The Les Shwab/Home Depot parking lot on Soscol is used by some but technically illegal. Walmart in Napa will not let you stay - parking lot is too small.
If you are nice people and don't like anywhere else give me a PM, I might let you park on my farm, 12 acres just outside of the City of Napa (about a mile from Skyline Park).