Hi Marc-
For the last two years, I've led groups of American high schoolers on pilgrimage, twice on the Camino (St Jean to Finisterre) and once on the Via Francigena (Lucca to Rome). We'll be back in Spain again this summer - it can definitely be done!
Galicia can be tricky in the summer, but June is much better than July or August. We've never made reservations and have always found beds, but it has been close at times. We do tend to maintain a faster pace than other pilgrims, though, and depart daily by 6am; if you would like a more leisurely pace, you might have more of a challenge. If that happens, though, you should be able to make reservations one day at a time, while walking. You can't reserve beds in most albergues, of course, but some of the private ones allow it and they advertise.
In my mind, the larger concern than finding beds is becoming part of the greater pilgrim community. Many pilgrims cast a nervous eye at groups of young people, but showing respect for the common behavioral expectations in the albergues, engaging in the generosity of spirit characteristic of most walking the Camino, and breaking into smaller groups whenever possible helps greatly with that.
It's a remarkable journey - and a remarkably taxing one. Walking the Camino alone is challenging enough; overcoming your individual pain while shepherding young people through theirs is a huge endeavor. Enjoy it, though. The relationships that you will be able to establish are well worth it.
Let me know if I can help you with anything else.