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?
One HUGE problem with your theory.
The law passed in the southern state that prompted the legal challenge, Mississippi I believe, predates covid by several years. Moreover, the anti-abortion movement spawned by Roe in 1973 predates covid by even longer. Casey (1992) was a challenge to Roe too, and that happened long before covid.
You're correct that as a consequence of Roe falling, should it, would be to open the door wider to SCOTUS saying federal vax mandates are acceptable. But they could do whatever they wanted on the issue, regardless of Roe.
Federal vax mandates would rely on Jacobson, an early 20th century case that allowed for forced smallpox vaccinations or you pay a fine.