In terms of vaccines and how they apply to the principle of individual rights, let's go back to what an individual right is. According to Ayn Rand, an individual right is a principle defining man's freedom of action in a social context. She correctly goes on to point out that the only fundamental right is the right to one's life in which life is a process of self-sustaining and self-generated action and, subsequently, people engage in activities to further and enhance their own life. The activities to enhance one's own life includes things like getting an education, doing things to stay healthy (ex: exercising or regular medical check ups), getting a job and maybe even starting a family.
She goes on to say that if physical force is to be barred from social relationships, men need an institution (i.e. government) charged with the task of protecting their rights under an objective code of rules. Now in the case of the anti-vaccine movement what they say is very revealing. Despite events like Influenza Pandemic of 1918 and how the flu virus at the time mutated into a deadlier strain killing hundreds of people, they would (most likely) ridicule such a notion despite evidence that refuted their claims about vaccines. Their attitude is very similar to Mary Mulligan (i.e. Typhoid Mary) who was quarantined for the rest of her life even though she promised health officials at the time she would not be a food service worker but did so anyway leading to her infecting many more people.
If an outbreak of a deadly disease that can spread quickly mandating or forcing vaccinations would proper. If people refuse to voluntarily be vaccinated despite being told the threat of the disease outbreak, either do it involuntarily if the person is healthy enough to carry the vaccine or quarantine them. This should only be done, however, as a last resort and it is determined by the CDC or US or state health departments where the outbreak has taken place, that there is not enough herd immunity and that the disease in question is serious enough to people's lives that such action is warranted. This is the same logic England used when it mandated vaccinations in Britain proper during a small pox outbreak that took place there during the 19th Century.
A proper government protects people's rights and in terms of vaccines and it is why when kids attend school they have to be vaccinated since unvaccinated children pose a threat not only to themselves but to the rights of their fellow students and the school faculty. There are states that have religious and conscience exemptions but it is looking like they will go away soon due to the increased cases of whooping cough, small pox and measles all three of which were nearly eradicated until the anti-vaccine movement emerged. Colorado, for example, is on the verge of removing vaccine exemptions due to outbreaks of whooping cough in the state in which young children are the most vulnerable.
For anyone to knowingly refuse to vaccinate oneself despite concrete evidence that vaccines help ward off contagious, life threatening diseases is tantamount to them saying they not only do not care about their life but also contributes to violating of the rights of others. This also includes participating in activities and disseminating information of groups such as the Vaccine Information Network whose information has been thoroughly debunked. The hatred of living if not human life itself ultimately is the basis of anti-vaccine groups. A logic no different than the environmentalist, anti-abortion and eugenics movements.