Do many academic institutions let students deposit their publications in the institutional repository?
Most university institutional repositories will allow students to submit their work, as the aim of an institutional repository is, generally, to capture the scholarly, research and creative output of a university. This would include work being created by faculty, of course, but also staff and students.

What differs between universities is the emphasis they place on different creators. At a large research university, increasing citation counts of faculty members will be a key consideration that may make the acquisition of faculty works a higher priority than student works. Liberal arts colleges will, based on their institutional missions as teaching centres, will be able to focus on undergraduate research on a level that many research universities would be unable to do.

I have recently become the Digital Repository Coordinator at a research university, and one of the challenges I face is trying to balance acquiring faculty and staff work vs. student work. Many of the faculty I have talked to, however, are excited by the potential of using the institutional repository as a way to showcase their students' work.

As a doctoral student, especially, I would assume that you would be able to submit work to your university's institutional repository. Looking at your profile, I believe that your university's repository does allow students to submit work (See http://digilib.gmu.edu/dspace/handle/1920/2883, for example). If it's not clear how to do so, contact your university's repository coordinator to make it happen! They'd more than likely be thrilled a student (1) knows what an institutional repository is, and (2) is interested in depositing their work.