So far my husband
@jungleboy and I have walked the Camino Francés and the Camino Primitivo as vegans and had no problems on either one. As
@Chaya said, there are lots of options on the Camino Francés especially. A surprising number of albergues on the route serve only vegetarian/vegan food, and we only came across one albergue that was not able to provide a vegan option for us.
The Camino Primitivo was much less populated so we had to plan ahead a bit more, but that was true for everyone, vegan or not. We did more self-catering on that Camino, but again, so did everyone. On several occasions we cooked and ate dinner together with a bunch of our fellow pilgrims, which was a lot of fun.
I write a vegan travel blog called The Nomadic Vegan, and I've published an article there with my top tips for eating vegan on the Camino. You can find it linked in my signature below.
While I haven't walked VdlP yet, I have traveled around the Extremadura quite a bit, so on my website you'll also find articles about vegan options in Seville, Mérida, Cáceres and other towns in the area. Those articles are a couple of years old, so I recommend using the HappyCow app to see what else has popped up since then.
In the south, don't pass up the chance to have the wonderful cold soups there -- gazpacho and salmorejo. Just ask them not to put ham or egg on top of the soup, as they often do.
Buen Camino!