On what grounds?
I don't know, I am trying to figure out which amendment covers them.
Well, before the (mis)interpretation of the 14th amendment, the federal constitution did not preclude states from enacting their own laws that might be more restrictive than those the federal government was allowed to impose.
Probably on the grounds that the Alabama State Constitutions allows for it. Given what I've read about the Alabama Constitution, it's quite likely, as there are a massive number of very specific Amendments to it.
In the absence of a U.S. Constitution word on the matter, the states, via their constitutions, can generally do whatever they want to.