You don't need a permit to purchase or own a firearm in Alabama, despite what the NYT might think.
You do need one to carry concealed, and it's against the rules of the University to bring a gun on campus, a student might be expelled and a faculty or staff member might be fired. But as always, those intent on doing murder and mayhem don't observe the rules.
But even if she had needed a permit, and been denied one, she was a Professor of BioChemistry, with access to all sort of chemicals and the knowledge to make "good" use of them. I'm sure she could have come up something even nastier. Bomb, chemical weapon, even a biological one.