I'm listening to a Rick Steves podcast where they're talking about staying in convents and monasteries, especially in Europe, but what I'm wondering is - is there a website / search tool for finding these accommodation sources?
I should have just kept listening - the interviewee (Trish Clark) went on to talk about the website which she's started:
Which bills itself as "An invaluable guide for travellers who want to stray off the well-trodden tourist path.", with the associated book: "Good Night and God Bless: A Guide to Convent and Monastery Accommodation in Europe".
This list is not complete, as I know there are more religious places you can stay overnight, but it is a start:
I have stayed in several Benedictine monasteries in the past, one of the group had written one of the monasteries in the country we wanted to go and asked about locations near our route. In one case we stayed in the hay loft of the monastery farm, low on hay due to the time of year, an other time I stayed in the hostel to the same monastery, out of season as only guest. Both times we stayed outside the actual monastery.
Most likely you will find them on internet these days.
Many monasteries do welcome people who want to come for a few days of quiet and contemplation but do not appreciate tourists who only come because staying is cheap.
One very popular one in that respect is Taizé, where you might be one of 3000 guests on an average summer week.