Ace and the Judge nailed it. Required medical procedures have never been imposed on the entire federal workforce. If Congress wants it, they are free to pass legislation.
There are areas in which the President's inherent Article 2 powers are inherently strong, areas in which the president is specifically named by the Constitution as having a unique role, areas in which logic and circumstances tell us he has a unique role -- commanding the armed forces, acting as the chief diplomatic officer of the US, stuff like that -- but anything that involves stuff that are easily legislated, and usually are legislated -- like requiring vaccination, or granting the power to require vaccinations -- are just not areas for asserting some previously-invisible font of Article 2 power.
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There are areas in which the President's inherent Article 2 powers are inherently strong, areas in which the president is specifically named by the Constitution as having a unique role, areas in which logic and circumstances tell us he has a unique role -- commanding the armed forces, acting as the chief diplomatic officer of the US, stuff like that -- but anything that involves stuff that are easily legislated, and usually are legislated -- like requiring vaccination, or granting the power to require vaccinations -- are just not areas for asserting some previously-invisible font of Article 2 power.