Yes, at Koya-san there's a central reservation office. You contact them, and they pick out your temple for you (there are dozens that take overnight guests.) Presumably they send gaijin to the ones that are prepared to take them.

The experience is like staying in a midlevel ryokan, except for the completely vegan food and the required service at 6AM. The temple I stayed at had lectures on Buddhism during the evening meal.

I and a young man from California were the only gaijin staying there, and he spoke no Japanese at all, so they put us in a dining room together and played an English-language video on Shingon Buddhism (the sect that is headquartered at Koya-san.)