The public health authority of the states derive from the police powers granted by their constitutions and reserved to them by the 10th Amendment to the U.S. Constitution. The basis for the federal governments’ authority to prescribe a quarantine and other health measures is based on the Commerce Clause, which gives Congress exclusive authority to regulate interstate and foreign commerce.
I can easily imagine the governors screaming like crazy if the Feds tried to take over local distribution of the vaccines.