Most private albergues take bikes, the advantage is you can ring ahead. The Albergue de Peregrinos in Irun takes cyclists, Islares, Aviles, Baamonde, Sobrado dox Monxes do as well, there is a probably a few more locally or association run ones who do as well. You usually have to wait in these non profit albergues until they are sure they will not have more walking pilgrims. Cyclists and walkers do share some of the route, but there is places where cyclists are recommended to take alterative ways, usually due to the Camino not being suitable for bikes.
I have not cycled it but know a few people who have, give yourself a lower km average for the Basque country (or to Bilbao).
I have only seen one cyclist come of their bike and that was the stage leading to Gijon, if you go that way be careful there is a rough forest track going downhill, she nearly went over the edge of a dropoff and she had to ground herself to stop it.
And
Buen Camino