There are a couple of nice campsites on the North shore of Kawaguchiko with showers and a place to get wood for a fire. I've also stayed at a good one along Ome kaido towards Okutamako. PM me if you want more details.
One of our friends biked through Japan, then Korea. He camped everywhere - check out the blog .. http://timsbiketour.com/ Japan is one of the most camping friendly places. Even if there are not official camp grounds, you can usually stealth camp with minimal hassles.
Yep have to agree with this - My team director and several other people I know have ridden Japan after graduating high school or Uni they hit the road and then camp where ever there is a safe spot to do so.
It’s a very common thing to do so I wouldn’t worry about official camp sites just find a nice spot and camp out.
Why campsites? There are many parks with toilets along the coast. Some michi no eki have showers and tatami style rooms open 24 hours. Wild camping lets you interact with local people and if you are lucky you can be invited over to spend a night.