Chiropractic is not illegal in Japan. There is no license/regulation for chiropractic so anyone can do spinal "manipulation". The numbers of practitioners are unknown, which means there can be no registration for it.

There are other modalities, two in particular, that are similar to some chiropractic methods which gives no reason to make another license under "chiropractic". However, it is a good advertising name so there are many clinics that use the term and thus, use chiropractic technique under their license. 

In Japan, only MDs or dentists are doctors. Unless the MDs say yes, nothing will be legalized.