Unlike some of the other disciplines such as ballet or swimming, there is very little information about gymnastics classes in Japan in English. According to Wendy Uchimura who kindly put this list together, in most of the schools the classes for younger kids seem quite easy-going, but once you hit the elementary school set, things get more serious and they have their own yearly competitions, as well as selected kids being able to start training towards national and international competitions.

Some terms which might help you are as follows:

Taiso kyogi = artistic gymnastics
Shintaiso = rhythmic gymnastics
Akuro = sports acrobatics
Kikaitaiso = apparatus gymnastics

Other related links (not classes):

Japan Gymnastics Association 

Shintaiso Federation (Japan Rhythmic Gymnastics Federation)

Japan Trampoline Assoction

Finally, check out Dance & Gymnastics Wear for leotards and other goods.