Armed Prohibited Persons Act of 2013 - Authorizes the Attorney General to make grants to states to develop and operate systems to retrieve firearms from armed prohibited persons.

Defines an "armed prohibited person" as a person whose possession of a lawfully purchased firearm became unlawful because of an act or omission committed by such person.

Requires a state's grant application to include assurances that: (1) the state's system will provide a reasonable opportunity for such a person to lawfully dispose of a firearm before the state takes any action to retrieve it, (2) the state will contribute pertinent information to the national instant criminal background check system, and (3) the state has a plan for the continued operation and maintenance of the system.