According to Wikipedia's article on the Siddhaṃ alphabet:
In Japan the writing of mantras and copying of Sutras using
the Siddhaṃ script is still practiced in the esoteric Buddhist
schools of Shingon and Tendai as well as in the syncretic sect of Shugendō.
A recent innovation is the writing of Japanese language slogans
on T-shirts using Bonji. Japanese Siddhaṃ has evolved from the
original script used to write sūtras and is now somewhat different
from the ancient script.
While your question focuses on modern use of Siddham in Japan, for a historic reference of it's introduction to China and Japan see Siddham in China and Japan - Sino-Platonic Papers. One of the things this paper shows is how Japanese is heavily derived from not just Chinese but from Siddham as well (somewhere around page 102).