Under the US Constitution, a public health authority could even make vaccination mandatory, and this was done in some historical epidemics.
In Jacobson v. Massachusetts, 197 U.S. 11 (1905) the US Supreme court held such mandatory vaccinations to be constitutional. The court wrote:
The liberty secured by the Constitution of the United States does not import an absolute right in each person to be at all times, and in all circumstances, wholly freed from restraint, nor is it an element in such liberty that one person, or a minority of persons residing in any community and enjoying the benefits of its local government, should have power to dominate the majority when supported in their action by the authority of the State.
It is within the police power of a State to enact a compulsory vaccination law, and it is for the legislature, and not for the courts, to determine in the first instance whether vaccination is or is not the best mode for the prevention of smallpox and the protection of the public health.
The highest court of Massachusetts not having held that the compulsory vaccination law of that State establishes the absolute rule that an adult must be vaccinated even if he is not a fit subject at the time or that vaccination would seriously injure his health or cause his death, this court holds that, as to an adult residing in the community, and a fit subject of vaccination, the statute is not invalid as in derogation of any of the rights of such person under the Fourteenth Amendment.
persons and property are subjected to all kinds of restraints and burdens, in order to secure the general comfort, health, and prosperity of the State, of the perfect right of the legislature to do which no question ever was, or upon acknowledged general principles ever can be, made so far as natural persons are concerned.
Railroad Co. v. Husen, 95 U. S. 465, 95 U. S. 471;
In Crowley v. Christensen, 137 U. S. 86, 137 U. S. 89, we said:
The possession and enjoyment of all rights are subject to such reasonable conditions as may be deemed by the governing authority of the country essential to the safety, health, peace, good order and morals of the community. Even liberty itself, the greatest of all rights, is not unrestricted license to act according to one's own will. It is only freedom from restraint under conditions essential to the equal enjoyment of the same right by others. It is then liberty regulated by law.
Whatever may be thought of the expediency of this statute, it cannot be affirmed to be, beyond question, in palpable conflict with the Constitution. Nor, in view of the methods employed to stamp out the disease of smallpox, can anyone confidently assert that the means prescribed by the State to that end has no real or substantial relation to the protection of the public health and the public safety.
The above decision was over a vaccination law during a smallpox epidemic, and the background is described in this "History Stories" article and in the Wikipedia article about the case
Governments have the authority, in general, to pass laws (or impose regulations) that serve "compelling governmental interests" and are of general applicability. Such laws are valid even over most constitutional claims, depending on the claim and the detailed facts.
Protecting the public health is a compelling governmental interest.
Such laws might have to (and normally do) provide exceptions for people with medical reasons why they cannot safely be vaccinated, or with sincere religious objections to vaccination. In such cases alternate measures of protection, such as frequent testing for infection, might be required.
Note that existing public health laws generally require wearing shoes and shirts in restaurants, and require restaurants to refuse service to those who do not comply.
In Compagnie Francaise de Navigation a Vapeur v. Louisiana Board of Health, 186 U.S. 380 (1902) the US Supreme Court upheld as constitutional an involuntary quarantine law.
in Zucht v. King, 260 U.S. 174 (1922) the US Supreme Court upheld as constitutional a public school district's exclusion of unvaccinated students.