How about Do-shi Village Camping sites?
There are several good camp spots I can recommend to you, which are located in the direction to West from the central part of Tokyo, or at the western edge of Kanto Plain as I live in the western part in Tokyo prefecture. One place you should definitely try, if you have not, is Do-shi Village Camping Sites, where you will find several camping sites available along the Do-shi river, the source of which stems from the Tanzawa mountains,which are one of the steepest high lands in Kanto Plain, meaning it'll provide really fresh air and pure water around the area, which you will definitely enjoy, and which you will find this is something that you are looking for when you are there!
How would you like those following Camp sites from the Do-shi Camping area?
Yamabushi Camping Site
Tsukiyono Camping Site
Hana No Moori Auto Campia