As a libertarian I believe in the rights of private parties to enact vaccine mandates. Why? Because vaccines work and private property rights are paramount. Libertarians accept that some societal rules that protect others would exist with or without government being the one doing the mandating. Speed limits, rules against harming others, insurance mandates, and other requirements would still exist in a free society. Responsible restaurants would still require its employees to wash their hands after using a bathroom even without the mandates of the health department. As a believer in strong private property rights, I believe that it would be irresponsible for a property manager to willingly disregard the safety of those using their property. Mandates are not against libertarian philosophy, especially if they would exist even without government coercion.
I would hope that hospitals, elder care facilities, and any facility that deals with at risk populations impose a vaccine mandate on their staff. Private and public universities should consider requiring the vaccine for students, just like they currently do for other vaccines. I would hope that when appropriate these mandates include exceptions for medical and sometimes religious reasons with potential alternate risk mitigation requirements like testing.
In 2016, Presidential candidate Gary Johnson stupidly suggested that Jewish bakers should be forced to bake a Nazi wedding cake. He was dead wrong about this from any libertarian angle you can find. Those who oppose allowing employers to set their own vaccination rules makes the same mistake. The owners of hospitals and/or their board of directors would be wise to require that their employees are vaccinated, even without a government order. Not having this mandate opens the hospital up for litigation.
At the same time, I believe in bodily autonomy. You have the right to choose to not be vaccinated. No one has a right to inject your body with any substance without your consent. However, with this choice comes responsibility. You do not have the right to endanger others. If you choose to not get vaccinated, you should choose personal responsibility. Isolate yourself as much as possible. Wear a mask in appropriate conditions. Respect the private property requests of others.
The moral dilemma are places that use force with mandatory attendance laws. This is true for prisons and government schools for (in NJ) those under 16. If you do not have everyone vaccinated the wardens of these two institutions are placing lives at risk. If you require vaccination the wardens are violating the bodily autonomy of their wards and their staff. In these cases, I see the problem not being vaccine mandates, rather the compulsion in attendance. We should work on alternative choices for both institutions.
The issue of mandates has been controversial among libertarians. The libertarian knee-jerk reaction has been to oppose mandates, however not all libertarians believe that in a free society we wouldn't still have privately required mandates. Walter Block has stated:
“My bottom line is that I think there could be circumstances in which compulsory vaccination would be required by law (let’s forget about who imposes them; well, governments for minarchists, private defense agencies for anarcho-capitalists) and parents should be required to vaccinate their kids. After all, people who do not vaccinate endanger not only themselves, but also the people they communicate with and thus break the NAP and should be punished accordingly.”
He posits that the COVID-19 threat has been greatly overrated. He may be correct here, especially now that doctors and hospitals are doing a better job at treating people who are infected. The number of deaths in many locations is down (but not all locations). The truth is that COVID-19 is a potential threat to both life and liberty. Perhaps, in some instances, more of a threat to liberty than life.