I suppose it is possble but the problem you might face is albergues being closed. This route is more isoltated and less travelled than others (which is its attaction as far as I'm concerned) but on the plus side the weather probably wouldn't be as severe as the CF-which goes pretty high is parts (ie the Pyrenees!). If you have the option may would be better as everything will be open but if you must go in the new year I think the VDLP is a good option as it starts low (25metres) and in the south where it would be warmer.
I was in the Galician section (read mountains) in July and it was cold then. I can only imagine Jan-Feb. And not sure on snow, visibility, especially arrows. Distances between towns? Sections where the cell phone did not have service. I am thinking safety here. Maybe someone in our forum who has done it in those months that had the experience? I loved the Via, and I am sure you will as well.
December and January are the real bottom-out months on ALL the Caminos. Nothing is open, and there are NO fellow pilgs to be seen some days... the Via de la Plata even more so. I´m sure you´d be safe enough, but you´d be quite lonesome! And wet. It RAINS down in Caceres all through winter.
Two years ago I set out from Seville on 2 January and walked to Salamanca where I had to return to the UK and complete the rest of the VdlP later in the year.
Whilst it is true that people were a little surprised to see me I had no difficulty finding accomodation. The albergues were open and the hospitaleros knew what was available up ahead.
The Amigos in Seville will be very helpful too.
At this time of the year this will be a remote walk - I only met one other pilgrim in 21 days - a guy who was driving to the start of each etapa and then getting the bus back to his car to g home. The VdlP in daily stages!
For many morning I had thick freezing fog - just me and the arrows like walking in a bubble. Then mid morning the sun would burn through and the temperature would rise. Almost like walking in summer.
With some planning, getting help if you need it phoning ahead etc - this can be a very rewarding experience.
I was walking on winter 2006 (late february) from Zamora. A lot of snow, even in he same Zamora city, and of course in Padornelo and A Canda mountains. Absolutely no pilgrims. All albergues for you. Different as albergues in Camino Frances, all you need is to know where to take the keys (in a paper in the albergue's door or asking to anyone). My camino is not usual, I had much more snow, rain and cold than normally.
I'm telling what I found, but don't be afraid, it's a wonderful Camino to enjoy.