Heidelberg University had options for the incoming international exchange students. However, I found my accommodation by myself via the Facebook group. I believe there are options for every budget, as there are different housing opportunities available. Most common way for students to live in Germany is to share a flat, or a WG (Wohngemeinshaft). This is usually the most affordable option (prices per room in such shared flat can be from 250 EUR/month or so), but the standards vary greatly.
My accommodation was located on campus, 10 min walk from my lab in BZH department, so this convenient location and the new type of the apartment were contributing to the higher price (400 EUR/month). I found the ad in following Facebook group: https://www.facebook.com/groups/wg.wohnungen.heidelberg/
also this one is useful: https://www.facebook.com/groups/wohnungoderwginheidelbergesucht/
I signed a contract with a girl (who was going away to do an internship herself) via Studierenwerk. She was officially subletting her room to me (I was a subtenant/Vermieter in German), and I found this way of acquiring housing to be very convenient.