Pity you will have to pass on the cafe con leche, but at least know that olive oil is the oil/grease of choice, not butter, so that will make things easier. I remember asking someone how many liters of oil he would go through in a year (single male) and he said 15!!!!!
You will not see meals listed as being vegan, but pork a la plancha, seved with fried potatoes and a bit of tomato being your typical pilgrim meal, you'll be ok. Same with espageti. Desert will be trickier as flan is made with milk (sorry you'll miss that), and yogourt is often offered, but so is fresh fruit. As for sandwiches (aka bocadillos) they seldom have anything in them other than the protein you have chosen: no butter, no nothing ;0) so easy to eat, if you can manage to eat that dry stale bread found in so many places.
Tarta de Santiago: any milk in there? Soooo gooood... I hope for you it does not.