Over the past seven years we have walked the Camino four times in winter (December/January), three on the Frances. It is my favorite time of year to walk. It is quiet and lends itself well to spiritual journey. Several years ago, when we started walking in winter we would only see 1 or 2 people each day. Last year we saw 10 - 20 people each day.
It does require a bit more equipment but it still fits into my small backpack. We have walked in 1 m of snow, whiteouts, and sleet storms. But only for a couple days and we could have laid back and taken a day off but we grew up in the mountains and so it seems familiar. Most of the time the weather is pleasant and the journey beautiful.
It does require more layers of clothes. We include -- a waterproof, windproof breathable shell jacket, down sweater, wool sweater, mitten shells and mittens (preferred over gloves), neck gaiter, warm hat, long thermal pants, waterproof/wind proof pants, and gaiters. I'm probably missing something but you get the idea. We also learned to carry a light sleeping bag as many places are cold in the winter and the heating is often sparing used.
Accommodations require planning and patience. And the days are much shorter. Albergues are not always available and rarely but occasionally a taxi is needed. But it has always worked.
Good luck! -- we'll be walking this December and January again and plan to arrive in Santiago de Compostela on/about January 3.