For sure in Santiago you should have no problem booking something for more than one night. Most albergues here are used to tourists and pilgrims staying for multiple nights.
Along the trail, as
@HighlandsHiker mentioned, it's usually the municipal albergues that only let you stay one night (understandably). However, having said that, I
have run into a private albergue that also needed everyone to leave by 8am - not in Santiago, but in Astorga. So...you may need to look into cheaper pensiones, hostales, and hotels instead of albergues if you want to be able to stay more than one night in certain cities. Just a note in case you didn't know that a
hostal is not the same as a youth hostel. A hostal usually has private rooms, whereas albergues usually have dormitories.
Booking.com is an excellent website and will filter choices for you by price if you like.
Buen Camino!
Faith