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. http://ace.mu.nu/
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.