What federal regulation gives the federal government the magical ability to mandate a medical procedure without consent?

OHSA, for employers subject to it. Medicare puts a lot of hospitals under their jurisdiction standards-wise, as well. Same for airlines. You'll note they've limited the mandate quite a bit to areas where there's wide court agreement the Feds have jurisdiction.

https://www.nytimes.com/2021/10/07/us/politics/biden-vaccine...

> Experts said that legal challenges to the rule were all but assured, but precedent is most likely on Mr. Biden’s side. In the past 20 years, “every standard that has been challenged in court has been upheld by federal judges,” David Michaels, a professor at the George Washington University School of Public Health who was a head of OSHA during the Obama administration, said in an interview.

> OSHA also has the authority to quickly issue a rule, known as an emergency temporary standard, if it can show that workers are exposed to grave danger and that the rule is necessary to address it. The rule must also be feasible for employers to enforce.

At the state level, there's plenty of legal precedent. https://en.wikipedia.org/wiki/Jacobson_v._Massachusetts