Right now pretty much everything is closed due to the lockdown order; the only legal camping currently available is primitive dispersed camping on BLM land. I can make a few recommendations for when the lockdown eases, though:
Lake Cachuma, in the Santa Ynez range north of Santa Barbara, is a large-ish campground with full hookups available. We used it for our first outing with the new van because there's a camp store, so we could fill in any items we didn't realize we needed. There are playgrounds and a disc golf course. Swimming in the lake is not allowed because it's a drinking water reservoir, but you can rent boats. Some sites have fire rings and some don't, so check when you make reservations if you plan to have a campfire.
Lake Casitas is similar, but additionally has a water park -- currently closed due to COVID, unfortunately.
Jalama Beach County Park is hard to get reservations for, but sometimes it's possible to snipe canceled reservations if you can go on short notice. It's right on the shore (to the point where I usually have to hose the salt spray off the van when I get back.)
Further north, Hearst San Simeon State Park is pretty nice, with a walking path to the ocean shore. It's easier to get into than some of the other state parks that are right on the ocean. (e.g. Morro Bay, Limekiln)
Any of these would make a good first camping experience, I think.
If you want something more primitive, the Los Padres National Forest has lots of campgrounds. I particularly recommend Figueroa, but it's not always accessible during the winter. These were pretty underused until the first wave of COVID restrictions, when camping suddenly became the only recreational option available to people. Almost none of them have hookups or dump stations. Most have pit toilets.
It's been a long time since I was 4.
But I think my main tip would be to not overthink it. Everything is new to a 4-year-old, so it's ALL interesting. When they get older maybe get a bike rack -- some of my fondest memories from camping as a kid are of riding around the campground on my bike.