Grades are ordinarily awarded by the teacher and reported or posted to whatever system, manual or digital, the institution uses for recording grades. My wife posts grades in the course she teaches to an online system.
I can think of no verb specific to the action of awarding final marks or grades in a course; what changes in this case is the verb's direct object, not the verb itself.
In my day (60s and 70s) many teachers also "posted", literally, a list of final grades on a bulletin board or on their office doors, but I don't think that is done much any more—nobody hangs around between semesters to consult such a list. In those days grades were mailed to students by the university during the break between semesters, but in the courses she takes my wife now learns her final grades at her online account.