It's already the law, and has been for like half a century (edit: more like 80 years, actually). If you aren't vaccinated you can't attend school. You can't serve in the military. You can't work in many health care fields. We requires our citizenry to be vaccinated against major preventable diseases. We always have.
And this policy is, objectively, the second[1] biggest success story in the last few centuries of public health policy. Period.
Why did no one care about "keeping federal powers in check" in 2019? Why do you only care now? You don't think maybe that there's something polluting your priors?
[1] The invention of antibiotics gets #1.