States and the Federal Gov fight over what is legal and illegal on a regular basis. Covid rules are normally only executive actions not legislative. It is subject to change rather quickly. The Supreme Court just recently overturned some federal mandates.(over simplification)
1. It doesn't matter what states and the federal government fight over. Florida can't sue the federal government to allow
its own vaccination rules to apply to border entry. States have no authority, constitutionally or statutorily, to determine border policy
in any way. See Article I of the Constitution.
2. Executive actions can be challenged, for sure. But national security and border policy is, again given to the federal government in the Constitution itself, and that authority is delegated to the federal agencies. Actions of federal agencies (e.g. that result from executive orders) performed under that legislatively and constitutionally delegated authority
is law. Law does not only mean legislation.
3. The Supreme Court overturned the broad mandate for vaccination for large private employers. They did not overturn it for everyone, like healthcare workers. And they did not rule that vaccine mandates broadly (e.g. requirements to be vaccinated to dine in a restaurant) cannot be implemented.
4. The Supreme Court is extremely unlikely to overturn vaccine mandates on
foreign nationals entering the country.