I think it is more common for students to register and vote on their campuses, but they are generally allowed to do either one. Absentee voting can be quite a hassle. The ballot needs to be requested well ahead of time. Some states or counties have strict rules about when/why you are allowed to vote absentee (although away at college is definitely a good excuse). Then the rules when you get the ballot can be quite convoluted. My D2’s first year away at college she voted absentee, and she needed to find another registered voter from her state to sign her ballot envelope as a witness (or find a notary)! She was way across the country from home, but fortunately knew another student from our state and they witnessed for each other. I just checked, and that is still the state law for absentee ballots from that state.
Voting rules are non-standard and arcane, and there have been big pushes in many areas in recent years to make it harder to get registered and stay registered in an attempt to try to suppress certain voting populations.