National Right-to-Carry Reciprocity Act of 2013 - Amends the federal criminal code to authorize a person who is carrying a valid, government-issued identification document containing that person's photograph and a valid permit to carry a concealed firearm in one state, and who is not prohibited from possessing, transporting, shipping, or receiving a firearm under federal law, to possess or carry a concealed handgun (other than a machine gun or destructive device) in another state in accordance with the restrictions of that state.