Roe V. Wade set several precedents. One of these legal precedents was to uphold the right to individual bodily autonomy in medical decisions. (obviously "bodily autonomy" doesn't apply if you consider the fetus to be a separate life. But that was not the court's take)
Back to the main point: They cannot mandate the vaccine if you have a right to bodily autonomy in medical decisions.
And even without a federal law like Roe, blue states would still keep abortion legal.
So I suspect the federal government is willing to sacrifice Roe V. Wade if it means they can mandate the vaccine.
Thoughts?
AFAIK Roe v Wade has yet to be cited in any vax case. It's a non-issue as far as vaccine mandates. The reason it's being challenged now is this is the first time in 40 years there's been a SC willing to overturn it. The reason it's happening now and not a year or two ago is because of how long it takes cases to worm their way through the appellate system to SCOTUS.