As the answer by Justin Cave correctly says, the US Federal courts will not hear a suit seeking to overturn a law as unconstitutional unless there is an actual case, with parties that are actually opposed in interest, brought by someone with standing, who is actually harmed or threatened to be harmed in a significant way by a law.
Moreover, since in In Jacobson v. Massachusetts, 197 U.S. 11 (1905) the US Supreme court held such mandatory vaccinations to be constitutional, no lower federal court can or will overturn that decision, even if it thinks the facts have changed or the prior decision was wrong. See State Oil Co. v. Khan, 522 U.S. 3, 20 (1997)
[d]espite . . . Albrecht’s "infirmities, [and] its increasingly wobbly, moth-eaten foundations," . . . [t]he Court of Appeals was correct in applying that principle despite disagreement with Albrecht, for it is this Court’s [SCOTUS] prerogative alone to overrule one of its precedents.
in National Coalition for Men v. Selective Service System No. 19-20272 (5th Cir. 2020-08-13), The Fifth Circuit wrote;
Here, as in State Oil Co., the factual underpinning of the controlling Supreme Court decision has changed, but that does not grant a court of appeals license to disregard or overrule that precedent. See also Roper v. Simmons, 543 U.S. 551, 594 (2005) (O’Connor, J., dissenting) (pointing out that only the Supreme Court may overrule its precedents
even where subsequent decisions or factual developments may appear to have ‘significantly undermined’ the rationale for [the] earlier holding
A plaintiff would have to bring such a case in a district court (or a state court), lose, and appeal, and hope the Supreme Court would agree to hear it and decide to overrule. With a very few exceptions, the Supreme Court is totally free to choose what cases it hears. It typically hears about 200and decides less than 100 cases per year (in recent years), out of several thousand cases it is requested to hear.