The short answer is yes, they can. In fact, colleges have been doing just that for so long that in most of the sources we have read, it's treated as a baseline assumption that it's not illegal for colleges to mandate vaccines. You likely remember having to submit proof of certain vaccines from your college or graduate school experience. Per Prof. Debbie Kaminer JD, a Professor in the Department of Law at the Zicklin School of Business, Baruch College (CUNY), "Many states have mandatory vaccination laws covering college students," and, "Private businesses have significant flexibility in adopting compulsory vaccination policies which, unlike government mandated policies, are simply not subject to constitutional restrictions."
So if the answer is yes, what about the timeline?
College students without preexisting conditions would tend to be later on the list of COVID-19 vaccine distribution and delivery. And we have never experienced anything in society like what this distribution will look like — think of the end of the movie "Contagion" where they use a ping pong ball-like lottery system. The Department of Defense will take the lead in operation Warp Speed, but they will deliver the vaccines to the states to disseminate. The states themselves then decide who gets the vaccine and when. Long story short, for the vast majority of college-age students — likely not until this summer. Which can be looked at in a glass half full or half empty kind of way. Next semester will still look extraordinarily unusual on college campuses, maybe even more so than this past semester (we'll do a podcast on that).
Should colleges require a vaccine? From the podcast below, the answer would seem to be yes. Dr. Paul Offit, who spent time in a polio ward as a young child and has dedicated his life to studying vaccines and the diseases that cause them, talks about the morbidity concern (which is exceptionally low in the typical college-age cohort) relative to the long-term, potentially lifelong vasculitis concern COVID-19 presents. For reasons still not quite known, COVID-19 can get in the bloodstream and affect almost any organ: kidney, liver, lungs, etc. And, with younger adults, cardiomyopathy has been a real issue. Hence, we almost had a canceled college football season — doctors were just realizing that fact. And this excludes the fact that college students with low morbidity concerns can still easily spread the highly contagious virus to people in different age and risk cohorts.
What's the good news? For starters, per Dr. Offit from a conference call with the CDC, historically, there has not been a single vaccine that presented significant side effects where those side effects were not present within the first 2 months. So we will know long before this summer both the efficacy and the safety level of the COVID-19 vaccine. Not all vaccines in history have been safe, but most have. The best example of this is the MMR vaccine for the measles – years of accumulated data show that it is essentially as benign as any intervention we know, and only requires one lifelong dose. Yet for a year period, a fraudulent doctor, Adam Wakefield, faked data in which he was able to publish a now completely debunked article that claimed MMR caused autism in children. And because his data was not investigated for several years, because he spoke so convincingly in the media, the great anti-vaccination movement got started. By the time he was completely debunked, it was too late to cease the movement. For more on this, there's a highly-acclaimed book, "The Doctor Who Fooled The World."
So the vaccine has better odds of being safe than I realized before researching this article, we should definitively know within a few months if that is the case, Dr. Offit predicts it is more likely needed to be taken once (like MMR) than every year (like the flu vaccine), and it would seem highly likely that all or almost all colleges and graduate schools will require it.
The bottom-line conclusion is that, barring some dramatic change — which with COVID is indeed possible; this virus has at times acted in truly unique ways versus any virus we have ever seen — college students will likely be vaccinated this summer, and college campus will look much more normal in Fall 2021.