Under what legal authority?
Arguably - just to play devil's advocate - under the theory that interstate travel is an activity that makes use of the instrumentalities of interstate commerce - i.e., traveling on the roads that cross state lines - and therefore since the federal government can regulate the use of the instrumentalities of interstate commerce, they can dictate who can, and who cannot, use them, and therefore they can condition use of the interstate roads on being vaccinated.
Of course, that argument really proves too much because the intrastate portions of the highways are also a component part of the instrumentalities of interstate commerce, and therefore if an individual uses an instrumentality of interstate commerce - i.e., a roadway whose right-of-way eventually crosses a state line, or an intrastate roadway that is necessary to the functioning of the interstate roadways - then that individual is also subject to regulation under the interstate commerce clause, and accordingly, under the interstate commerce clause, the federal government can mandate vaccines for any person who travels anywhere other than on his/her own private driveway.
