A recently published working paper by Riley Acton of the University of Miami, Wenjia Cao (Michigan State University), Emily E. Cook (Tulane University), Scott A. Imberman (Michigan State University), and Michael F. Lovenheim (Cornell University), indicates yes they did. The study found that "college vaccine mandates led to 5.4 fewer deaths per 100,000 in the county where the college was located." An August 2, 2022 Forbes article about the study can be found here https://www.forbes.com/sites/michaeltnietzel/2022/08/02/college-vaccine-mandates-saved-lives-in-surrounding-community-according-to-new-study/?sh=708d25df5762.
Not surprisingly the reopening of college campuses in the fall of 2020 helped increase the spread of Covid-19. A new peer-reviewed study released last month found that “reopening a college was associated with a statistically significant increase of 4.9 cases per 100,000.” However, the working paper by Martin Andersen, Ana Bento, Anirban Basu, Christopher Masicano, and Kosali Simon, also found that “hospitalizations, ICU admissions, and deaths did not increase, on average.” An article about that study can be found here https://www.insidehighered.com/quicktakes/2022/08/30/study-college-reopenings-increased-community-covid-19-cases.
Trying to find an anecdotal connection between these studies is both interesting and challenging. However it seems logical that if living near a college campus made it statistically more likely that you would become infected with Covid-19 back in 2020 when vaccines were not available; and if becoming infected with the virus in those cases did not increase the likelihood that you would be hospitalized or die; then when vaccines became mandated by college's and universities deaths from the virus would decline.
That said, interpreting the results of one study without the benefit of the other would lead some to believe that without the vaccine mandates imposed by most colleges and universities deaths from Covid-19 in communities near campuses would have increased. However, the findings of the study by Anderson et al referenced above, "College openings in the United States increase mobility and COVID-19 incidence" https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0272820, seems to dispel that theory.
There is no doubt that vaccine mandates that were originally implemented as a way to keep a campus safe from the virus, resulted in a positive collateral impact on the surrounding community in regards to deaths from the virus. However, what is not clear is what the death rates would have been if the mandates had not been implemented.