The "Stay at University Housing" idea seems to be a popular one among some "budget travel" writers, because it is an idea that I see from time to time when reading articles. On some level, it seems like it could make sense - colleges have lots of empty rooms over the summer - but on the practical level, it just doesn't happen. Universities generally go into barebones staffing when classes aren't in session, and dorms are generally shut down. They might keep one or two buildings open for summer/international students, and perhaps have rooms available for people taking part in summer camps, conferences, or other activities on campus, but other buildings are generally locked down. They just aren't in the business of short term rentals, and it isn't worth the hassle it would take to staff a dorm like a hotel.

There might be some exceptions, but they are few and far between. You would have to contact each individual university to see what their policies might be, and as Buck indicated, its likely going to be a very time consuming, and fruitless task.

I'd recommend sticking to other budget options, like hostels, budget hotels, or camping.