Booking Ryokan

What is nowadays the best way to book a Ryokan?
How long in advance do Ryokans generally take reservations?

There is no best way, each option has its pros and cons.

Aggregators websites (Booking, Agoda, Jalan, Japanican, Rakuten...):
+ sometimes offer huge discounts; more user friendly (no need to enter your name in kanas, for example)
- less options (especially for meals) and room description is lacking

Direct booking:
+ has a lot more variety of "plans" to choose from, which are described in detail (regular meals/bigger meals/specialty meals/discount for late check-in, early check-in, late check out, gifts in room, etc). Also easier to request for additional services, like a shuttle to the hotel. Some ryokans only offer their best (and more expensive) rooms to direct customers.
- usually more expensive

Basically, if you want to know exactly what you are booking, don't use aggregator websites.

I  usually book directly from their websites. On several occasions their website said no vacancies for the days we wanted, however, we still managed to book through third party websites, albeit more expensive. 
An  unusual experience  in September last year, we were trying to reserve a room in The Lake Suite Ko No Sumika for  November, and  regardless of how hard we tried the website just refused to accept our reservation. Finally we emailed the hotel and thought that the website could be the problem. Interestingly, we received a reply mentioning that  a direct reservation was suspended , and were advised to try a third party for a reservation, but they were prohibited to name which company to make a reservation from. We had to search several websites, and eventually managed to reserve through a Japanese website, again the rate was unusually high.