@sembry  

You asked your question some time ago, so I'd be really interested to hear your experience!  So please do come back and share what you learned!

There were a couple different ways I achieved this when I was teaching.  Sometimes I elected to use Collaborations.  I assigned a collaboration to a group, or an individual student, and I was able to view every transaction on the google doc, as well as have students submit them to the assignment.

In other times I had students create a google doc and submit it either directly from Google Drive or as a URL.   @jperkins  did a great short post on Grade a Google Doc using SpeedGrader.  He would have more expertise in accessing the history than I would, so hopefully he will add his experience!