I worked customer support for Blizzard Europe for several years until about 2012. At the time they ran their own infrastructure co-located in ISP data centres. It was very important to Blizzard to have as much control of the quality of the user gaming experience as possible, with control of their infrastructure being a large part of this, and I would be very surprised to hear this philosophy of theirs has changed.
Then again, I don’t know for sure what they’re doing these days.