A member of a volunteer Brooklyn security force that works closely with city police was charged Monday for allegedly paying officers in the New York Police Department to expedite gun-permit applications, the latest development in a burgeoning federal corruption probe.

Alex “Shaya” Lichtenstein was charged with one count each of bribery and conspiracy as part of a three-year scheme in which he claimed to have secured more than 150 gun permits through bribes to officers, including for people who otherwise might have been rejected...