In an ideal anarchist society, there would be no one who could legitimately mandate vaccinations. If there were some individual or group that took it upon themselves to pay men with guns and blue uniforms to kidnap and cage people who refused to vaccinate, the vast majority would recognize this as criminal behavior. The people would be entirely justified in taking up arms against such an individual or group. The widespread recognition that using force in this way is illegitimate represents the essential difference between an anarchist society and a statist one.

Now just because vaccinations can't be mandated, doesn't mean they can't be incentivized. Health insurance companies and doctors could refuse to insure or treat the unvaccinated and their families, or only do so at much higher costs. They could also offer the vaccinations free of charge. Why would they do these things? Because, if the vaccinations truly beneficial, it would be cheaper to insure and treat these clients, and they would have to pass these savings along to the clients in order to attract them. If they did not, competitors could offer better deals on price and service, and steal away these more preferable clients. Eventually this would become universal, as it would be unprofitable to do otherwise.

In this way, competition in the free market aligns the interests of the companies and their customers. When your insurance company requires vaccination, it does so in its own self interest. But because clients have a choice to go elsewhere, the client gets the benefits as well. When a government mandates vaccination, it is not always clear whose interests are being served, and we get these conspiracy theories.

In addition to health care providers, other organizations could have similar practices that incentivize vaccination. Schools, which would all be private, could refuse unvaccinated students. Imagine a housing developer who builds a neighborhood, and maintains the roads, sewers, and other common areas. When that developer sells a new home in that neighborhood, as one of the provisions in the contract of sale, it could have a requirement that all residents must be vaccinated, and that in order to resell the property, your buyer must also accept this same contract. They would have such provisions in order to make it easier to sell the other homes in the neighborhood.

We all see the web that holds a statist society together. But when many imagine a free society, they imagine that web gone. They cannot imagine society functioning without it. But anarchists are not advocating the removal of society's web, just its replacement with one based on voluntary interactions rather than forceful ones.