I didn't walk with a dog but with a cat. It's possible but not easy.
Most accommodations do not allow pets. Albergues, hotels, pensiones, doesn't matter. It's getting better though.
Some do allow them now, luckily. So, you'd need to plan carefully and make reservations beforehand.
I saw several people walking with their dogs. Mainly spaniards on the last 100kms. Met some on the campsites.
One american pilgrim I remember who brought his dog. Dog was extremely well trained and it all worked out, but the owner had to make many phonecalls every day to find accommodation.
I sometimes got accepted into places that usually don't allow pets, out of pity. But that's nothing one should plan for (I found the cat on the route, so had to improvise). I was lucky to have a tent. The campsites in Sarria, Portomarin, Arzua and Santiago were very pet friendly.
With a dog, I'd plan for very short stages (not more kms than you walk with them per day at home). If camping is not an option, book your accommodation before.
Travel to / from Spain can also be a problem. Dogs over a certain size are not allowed on trains (that might have changed recently with the new animal welfare laws, not sure). So you also need good planning for getting there / back home.