Thank you everyone for your ideas. It looks like the solution is to have the students export their programs to an AIA file, manually send it to me through some system (I'm thinking a Google Form with file upload), then have me import it for grading. The naming standard is a great addon to that idea.
It's rather cumbersome to go through all that for each project for each student, but it's what I was doing before with the previous system I used to teach coding (Alice). I was hoping that since App Inventor is all online there'd be a way for them to simply share their programs with me through a link or some such.
Thanks to who mentioned Google Sheets as a feedback system. I have found it incredibly useful for creating marking schemes and feedback systems over the years! I find it the best part of GSuite, and use it whenever possible.