There is a fair amount of free or inexpensive lodging available on the trail. Sometimes that can mean sleeping in a hut with walls that only go half way up to the ceiling, other times it's simple rudamentary lodging, sometimes it's at a temple. It's nothing like a paid minshuku or ryokan, and can be pretty spartan sometimes, but hey.... it is free or cheap.
Here are the English resources i know of:
The most famous list is the Japanese list called the Hagimori List, after the man who puts it together. I only have it in .jpg format because that is how it was sent to me. Here is >>Page 1<< and >>Page 2<< of the most up-to-date 2017 list.
An organization called O-Henro House Shikoku 88 Project, or Henro House in short, has recently started a project of converting older, unused building into inexpensive lodging for walking henro. As of now (July 2017) they have 20 lodges in operation. They say it is "membership based," which means you need to register with them before you can use any of the lodgings, but there is no cost to register. Reservations are required before you can stay and the cost per night is ¥2,800 with no meals included.
Call for more information