Haven't seen a belt-fed FA shotgun but there have been FA shotguns built; both conversions ( FA Remington01187 comes to mind) and purpose-built (Atchisson, USAS 12 etc). Shotgun News in years past had a regular advertiser that sold plans for a belt-fed conversion for some shotugn, don't recall which one.
The book The World's Fighting Shotguns by Thomas Swearengen has a chapter on FA shotguns that is very interesting.