Can anyone recommend a hiking and/or biking tour operator for Japan?

Many thanks.

japan-guide.com is almost always a good place to begin any research on Japan.

https://www.japan-guide.com/e/e2427.html
https://www.japan-guide.com/e/e2210.html

I have wanted to hike the Shikoku Henro Trail. There are 88 temples on the trail, but you don't have to do the entire trail. You can sleep in the temples on the way. Perhaps you can look into it. You don't need to go on a tour.

You really don't need any tour. For the hiking though, you need to clarify if you mean real hiking out in the boonies, or various day hikes where you can still have some nice or even luxurious accommodations for your overnight. One of the better sources of hiking in Japan is naturally called Hiking in Japan. Japan is around 70% mountainous, so there is no shortage of places for hiking regardless of where you go. Near Tokyo, Okutama and Takao are well established and with good infrastructure. For Kyoto, some go up to Kibune and Kurama, or Takao (a different one), Nagano is superb for its mountains and hiking. You just need to decide where in Japan you plan to go to.