Sabika Sheikh Firearm Licensing and Registration Act This bill establishes a process for the licensing and registration of firearms. It also prohibits the possession of certain ammunition and large capacity ammunition feeding devices. First, the Bureau of Alcohol, Tobacco, Firearms and Explosives must establish (1) a licensing system for the possession of firearms or ammunition, and (2) a registration system for firearms. In addition, the Department of Justice (DOJ) must establish and maintain a publicly available database of all registered firearms. Next, the bill creates licensing requirements for the possession of a firearm and ammunition. DOJ shall issue such a license if the individual is 21 years of age or older, undergoes a criminal background check and psychological evaluation, completes a certified training course, and has an insurance policy. It also outlines the circumstances under which DOJ must deny a license (e.g., the individual was hospitalized with a mental illness). It also establishes additional requirements for an antique firearm display license and a military-style weapons license. The bill generally prohibits and penalizes the possession of a firearm or ammunition unless the individual complies with licensing and registration requirements. Further, it prohibits the transfer of a firearm or ammunition to an unlicensed person. Finally, it generally prohibits and penalizes (1) the possession of ammunition that is 0.50 caliber or greater, and (2) the possession of a large capacity ammunition feed device.