Seconding the recommendation to use a Japanese booking site. I use Jalan.net, I find their layout more intuitive than Rakuten which is the other big website.

Once you have found a ryokan you like, compare the staying plan and the price versus the ryokan's official website and the western hotel search sites as sometimes one is cheaper than the others or has a better selection of rooms and dining plans.
For example, I am very familiar with western affiliate cashback sites (~5-7% depending on site and valuation) but do not travel enough to want to deal with Jalan points, so if everything is the same then I will book via booking.com.

Then book.

Friendly reminder to make sure you are using the Japanese version of these websites. Use Google Translate if you need to - but the selection can be quite different from the English website.