Alabama is a May Issue State and only issues permits to residents of Alabama. May Issue means that the sheriff may or may not issue you a permit even after meeting all requirements for an Alabama Pistol Permit.
Here is the exact text from the Alabama Firearms and Weapons Law:
§ 13A-11-75. License to carry pistol in vehicle or concealed on person –
Issuance; term; form; fee; revocation.
The sheriff of a county, upon the application of any person residing in that county, may issue a qualified or unlimited license to such person to carry a pistol in a vehicle or concealed on or about his person within this state for not more than one year from date of issue, if it appears that the applicant has good reason to fear injury to his person or property or has any other proper reason for carrying a pistol, and that he is a suitable person to be so licensed.