How to book Ryokan inn that don’t have website?

There an inn called Shitanda in Takayama that I would like to book, but using every hotel site it says all the date are not free even if I go out 2 years. I found this to be the case with many other inns.

Is the only way to book to call up, I have sent an email but it was written in English.

Please advise on the best way to book such hotels for a Japan newbie.

Thank you.