Thanks again everyone for all of the help! We ended up camping at Laguna Seca. There are hot showers and flush toilets, and the sites are nice. Facilities are clean (though my wife was a bit spooked by the lizards lounging in the bathrooms!) Not much privacy, and the campgrounds are all on fairly steep hillsides, but on a Friday night in early June it was maybe 10% full. Some sites have shade, most are in the sun. Many are up on a ridge and have beautiful views over the track and/or the valleys around. We paid the extra $5 for a site with electrical, only because it was in a nicer loop of the campground. $35 for the night.
Overall: not a place I'd travel to stay at, but really very pleasant as a staging point for Monterey and thereabouts.
Highlight for the kids: the driving school started up around 10am in the morning, and started the day by driving the students around the racing line in vans. So we got to watch three E350 EBs racing at Laguna Seca!!