You are in luck, there is an excellent book that is exactly what you are looking for called "The Other Side of Zen: A Social History of Sōtō Zen Buddhism in Tokugawa Japan" by Duncan Ryuken Williams.

It focuses on Soto Zen Buddhism but this was by far the most common form of Buddhism at the time and also was highly influential with other forms of Buddhism. For example, the Soto Zen style funeral ceremonies became standard across all sects.

One thing you'll encounter in that book is the Danka system which was a mandatory family registration at a local temple. This was the engine driving the ubiquitous existence of temples in Japanese society. Check out the citations on the "Danka" wikipedia page for some more reading on that topic.

If you are looking for Buddhism in more recent Japanese society check out "Japanese Rinzai Zen Buddhism Myōshinji, a living religion" By Jørn Borup. I'd say buy it new to support the author but only used copies are available so check out the PDF (https://terebess.hu/zen/Myoshinji.pdf)

It would also be helpful in that much of how Rinzai zen is now run does not differ too much from how it was run in the Tokugawa period.