Well, it's certainly completely possible to build a system that would do what you're looking for, but I'd like to believe you can find something off-the-shelf that would accomplish it, since a custom system would definitely cost you more (e.g. thousands of dollars to get a decent system built by a reputable firm).

I'm guessing that the students are the clients in this setup?
One thing to watch out for is that if the student/client includes their name and/or contact information on their essay, then you (or the system) would ideally sanitize that information out of the essay before sending to the the teacher/employee.

I don't know a system that I'd recommend for this, but I hope that someone else is able to recommend something that you like.