Does the the "privacy" protection afforded by the Bill of Rights covers unwanted medical treatments, including unwanted drug taking, notably mandatory vaccines?
It probably does, up to a point. Roe v. Wade asserts a right to privacy, discussed in §VIII. Granting that there is no explicit enumeration of a right to privacy in the Constitution, its implicit presence is discerned via a long series of constitutional rulings of a diverse nature. It is not clear what is the extent of
This right of privacy, whether it be founded in the Fourteenth Amendment's concept of personal liberty and restrictions upon state action, as we feel it is, or, as the District Court determined, in the Ninth Amendment's reservation of rights to the people...
However, even in the case of explicitly recognized rights, they are not absolute: you cannot commit fraud or threaten a person with death and escape punishment by citing the 1st Amendment, you cannot own a machine gun and cite the 2nd in your defense. Fundamental rights are strongly protected, but they may be limited in a fashion that survives strict scrutiny. This means that the encroachment is necessary to a "compelling state interest", it is "narrowly tailored" towards that end, and is the "least restrictive means" to achieve that end.
The question arose in Jacobson v. Massachusetts, 197 U. S. 11 where Jacobson was criminally arraigned for refusing to comply with a mandatory vaccination law (applicable to all persons over 21). The court noted that
the liberty secured by the Constitution of the United States to every person within its jurisdiction does not import an absolute right in each person to be, at all times and in all circumstances, wholly freed from restraint. There are manifold restraints to which every person is necessarily subject for the common good.
The liberty secured by the 14th Amendment, this court has said, consists, in part, in the right of a person 'to live and work where he will'...; and yet he may be compelled, by force if need be, against his will and without regard to his personal wishes or his pecuniary interests, or even his religious or political convictions, to take his place in the ranks of the army of his country, and risk the chance of being shot down in its defense...
According to settled principles, the police power of a state must be held to embrace, at least, such reasonable regulations established directly by legislative enactment as will protect the public health and the public safety.
The right to compel vaccination is reaffirmed in Zucht v. King, 260 U.S. 174.
There is currently no mandatory vaccination law applicable to adults; were such a law to be created (analogous to the earlier Mass. law regarding smallpox vaccination), it could easily pass judicial review as long as it is "minimalist". The question of "compelling government interest" would distinguish between mandatory Ebola or zombie-fever vaccinations vs. shingles or (ordinary) flu. School-related vaccination laws are the most minimal way to achieve the desired outcome, so a law requiring everybody to submit might not pass a strict scrutiny review.