Kev-
I've led student groups of around 10 people on the Camino Frances three times before and, for the most part, have had no trouble with the albergues accommodating us. The only one that outright refused admission to groups was the albergue in Palas do Rei.
I'm gearing up for a student trip on the Norte/Primitivo this June/July and was a little more nervous, given that most of the albergues top out around 20-25 beds, and often have less. Where possible, I've emailed albergues and been told that they are fine with groups, provided beds are available. Reservations, of course, are not an option in the majority of albergues.
I think you'll be fine on the Primitivo in April, but you might have a few people bring sleeping mats, just in case.
Dave