Camino de Santiago in Winter

I have no experience with winter travel in Europe (brrrr), but if you're interested in commercial accommodations, you can use a site like booking.com to see how many places have rooms for the period you might visit. Try a few cities and towns to get an idea. I assume you'll be OK in major cities like Burgos, Leon, or Santiago de Compostela. The small places don't have a lot of lodging options to begin with, so if just one or two were closed, it would make a real dent in your choices, but then again, you might not have much competition from other travelers.

If you're looking for more casual places like pilgrims' refuges, especially in small towns, I can only suggest that you Google for Camino de Santiago discussion boards and post an inquiry there if no one here can help you. For small towns it would also be good to have an idea of what you'll find in the way of food.

I'd be surprised if significant sights were closed in January, though it's common for hours of operation to be cut back, and I suppose something might be closed one or two days a week during the off-season.

I trust that your guide book mentioned that one reason things will be less crowded is that the weather makes winter a less-attractive time to do the walk. Do check on historical weather patterns. Part of the Camino de Santiago runs through the mountains, and from the Basque Country to Galicia you'll probably encounter rather chilly weather and some rain.

Wunderground.com has historical day-by-day weather data going back at least a few years (it varies by city). If you select a date in January and choose "Monthly", you'll be able to scroll down to a graph that shows the temperature variation over the entire month of January for whichever year(s) you choose to look at.

There may be some stretches toward the eastern end of the walk where the weather would be warmer.

this is the go to Camino forum

https://www.caminodesantiago.me/community/forums/camino-frances.12/

In January you will encounter snow and ice, which sometimes prove fatal.

Spain had a lot of snow last year and many routes are getting snow already. A German peregrino just died from slipping and falling the other day.
Visit the Camino forum. It is a treasure trove of info.