OK, we survived. The camp site was in the mountains near Hadano, about 1.5 hours from Tokyo by car.
Here on Google MapsThe trip to the campground was straightforward though the last 5 or 10 miles are very twisty and the road narrows to single lane fairly often. We passed a lot of cyclists going up the mountain. I like cycling, but I much prefer the car for climbing mountains.
The campground is set up for car campers and is set in a valley with a small river running through it. Lots of Japanese pines and other trees.

We got the tents set up without any issue.

Then we all took a moment to enjoy the scenery. In this picture, Mrs Putaro, Princess Pyromania (6 - more on the pyromania later) and YT. My son, Bug Boy (8), is behind the camera.

We then went for a walk to see where the river went to.

The river hit a drop-off less than a hundred meters away from our tents. (That's Bug Boy in the picture). Note the very small "Don't Go Off Here Or You'll Die" sign.

That looks to be a good 10 meter drop. Guess we'd better put the inner tubes away!

After dinner we were going to have a campfire but it started raining and we got a couple hours of heavy rain. We put the kids to sleep and turned in early ourselves. Around 10 PM the rain had stopped, I was asleep, and a small voice came from outside "Dad - I have to go to the bathroom" so I took the Princess over to the toilets (note: these were the BEST campground toilets ever - clean and they even had those "washlets"). Afterwards she didn't want to go back to sleep and the rain had stopped so she and I sat up and started a campfire - fortunately I had bagged the wood when it started raining. We didn't have much kindling, though, so I finally resorted to getting it started with the gas stove. So much for wilderness survival skills. She enjoyed adding wood to the fire (hence the "pyro" designation) and we burned through our little pile of wood and I finally got her to fall asleep around midnight.
The next day we had a traditional camping breakfast of bacon & eggs and "Bug Boy" found a number of insects to chase around and torment in addition to his sister. We broke camp and got rolling back down the mountain by around 11 AM. At the entrance to the expressway is a very nice onsen (public bath) where we stopped, all had a shower and a soak in the hot bath and then some ice cream afterwards and I got 10 minutes in the massage chair. Then we grabbed some lunch and headed back to Tokyo.
So, nothing terribly adventurous. This was the first camping trip with the kids and the first one we've done since we've been in Japan and it worked out well. Hopefully we'll be able to do more.