Hi Jakke,
The last time I completed the Frances, there was one Spanish pilgrim I met that brought his boxer. I spoke with him in depth as to how his pilgrimage was going, and I must sadly say that it was very difficult for him, and especially his dog. Most albergues turned him and his dog away because they did not want to deal with the liability/responsibility of having an animal on the property. So he had multiple 40+ km walking days just to find lodging that would accept both of them, and he believed that most of the albergues that did let him stay only did once they saw how exhausted he and his dog looked.
He said the albergues that did let him stay were a mix of private and "public" albergues, though none of them officially allowed pets, but were more willing to make exceptions for service animals, and some let him stay at the albergue because a resident of the village/town he was in would let him keep the dog at their house/yard. He even mentioned that one night he stayed in a barn with his dog. He also mentioned that accommodations that accepted horses would take the two in, but they were pricey, and the distances between these places was quite far, unless of course you were going by horse.
As for his dog, I recall seeing it twice. I met it first in on the road in Rioja, and the dog was as happy and cheerful as anyone would expect a dog would be. Then I met it in an albergue outside of Burgos. That time, I remember it being so exhausted, it did not move from its bed all night, and I don't think it didn't have the strength to even eat anything. The owner was concerned and he said he was planning to take it to a vet the next day. We made it to Burgos the next day and I did not see the pilgrim, but one of the other pilgrims did have his contact info and told us that the vet said the dog was very ill and could not go further, so they both went home back to Madrid.
As for vets on the Camino, the pilgrim did mention that he had vets check his dog multiple times while they were on the Camino. He didn't mention having a list, but he did say that all he did was ask the host at the albergue how to find a local vet, and he never had trouble getting one.
I hope someday that the Camino will be more dog friendly, and perhaps someone else will have a better story to tell about this subject.