If it were me, I would steam everything as a precaution. The mattress, box spring, frame, bedding, all of it. If everything gets up to 180 or 200F, it'll kill any bugs or eggs.
It is entirely possible that they have them at the school, you should probably let them know that your son is coming home with bites and see if they'll treat the classroom.
If the bites are mostly on his back, then they might not be bedbugs. Bedbugs usually travel along veins where they are close to the skin (arms, legs, neck, etc) snacking along the way. You would see a trail of hard red bumps almost exactly on top of veins.
Sarcoptic mites (scabies) on the other hand, are nearly invisible in bedding and clothing, and can bite anywhere. They burrow and lay their eggs under the skin, so it takes a couple of days to a couple of weeks before the rash shows up. It's exactly the kind of thing that one can get from a grungy rug. (Not to mention public transit and office furniture...)
The one thing in common is that steam can kill anything, so the same advice about steaming the bedroom (and the living room, couches, chairs, etc) should take care of anything you've got once you get the people in your household treated.
Take him to a dermatologist if you can, and be sure to mention it might be something he brought home from school.