Can you "Hear back"?

Solution 1:

Both are OK, but they have slightly different meanings. In the case of heard back from, there is the implication that the speaker had spoken/written to the person already, and was expecting a response. In the case of heard from, it simply means that the speaker was expecting a call from the person, not necessarily that there had been an earlier communication to which they were expecting a response.

So heard back from is somewhat more restrictive.

Solution 2:

The back means "in response to a previous communication". The Ngram viewer finds a use from US News & World Report from 1955, so people have been hearing back from others for some time now.