Since only one or two of the accommodations are geared to pilgrims, and since there are not many pilgrims, whether or not places are closed won’t have anything to do with pilgrim traffic, or the lack of it. Private accommodations are likely going to be open year round, except for the tourist-specific places, such as some of the places in As Médulas. But I have gotten confirmation from both the Complejo Rural Agoga and Casa Socorro that they are open year round in As Médulas.
If you search through the old Invierno threads, you will see that a couple of forum members have walked in the dead of winter,
@chinasky being one of them.
In terms of safety, of course bad things can happen anywhere, but I have walked the Invierno three times now, once with a companion but only for a few days, and as a solo female had no problems. Quite the contrary, people were oh so kind and hospitable. The forum has a great little guide, which is here in case you haven’t seen it:
https://www.caminodesantiago.me/community/resources/invierno-forum-guide.535/
And if you do use it, it’d be great if you would post back with comments and suggestions, because we will be doing an update in the fall or winter. Buen camino, Laurie