Poking around with some searches, here is some starting information to get you going.
Mangels Shooting Gallery, located at 1214 Surf Ave, Brooklyn, NY. Opened in 2013.
Dick Zigun, founder of The Seashore, Coney Island, and the one responsible (apparently) for resurrecting a shooting gallery at Coney Island.
Rhiannon Shaefer, great-granddaughter of William F. Mangels, the original gallery designer.
Charles Denson -- Coney Island History Project director. Uncertain what organization this guy works for, or how to contact him.
Phil Schreier, Senior Curator, NRA National Firearms Museum.
Coney Island Shooting Gallery from 1940s Makes Comeback
The NRA National Firearms Museum also has some information. Senior Curator Phil Schreier. "Restored carnival attraction had museum built around it. On today's Curator's Corner, John Popp speaks with National Firearms Museum Senior Curator Phil Schreier about a shooting gallery, made by the F.W. Mengels Company in 1903 as a steam-powered contraption and converted to electricity in 1918. The gallery features rows of moving ducks, which competitors shoot at using special .22-caliber rounds designed to crumble upon impact. The National Firearms Museum performed major restoration on the gallery before sealing it in a room and constructing the museum facility around it, meaning that the exhibit can only leave if the building is torn down.
Wikipedia has scant information on William F. Mangels
Dick Zigun Unveils Restored William F. Mangels Shooting Gallery @ Brooklyn Daily
Information on the original galleries designed by Mangels ...
Antiques & Collectibles - Lifestyle - The Boston Globe
A cast-iron shooting gallery made in a small Coney Island, Brooklyn, N.Y., machine shop in the late 1800s was the top seller at Grogan & Co.’s auction of the Elli Buk Collection of scientific, medical, and technological artifacts. Buk, an antiques and art dealer in Manhattan’s SoHo neighborhood, died last year at 62.
The gallery comprising over 150 bird, rabbit, and other targets, sold for $60,000 against a $5,000-$10,000 estimate. It was made by William F. Mangels, who in 1883, at 16, immigrated to this country from Germany and became known as “The Wizard of Eighth Street” for his amusement park inventions, including The Whip, The Tickler and the jumping carousel horse.
A news story about the opening of the new shooting gallery:
World War II-Era Shooting Gallery Returns To Coney Island, by Willie Simpson on Aug 7th, 2013
Some information about restoration at the Oakwood Amusement Park, which is apparently built up at the spot where the old amusements at Coney Island existed. At this spot, numerous old artifacts from the era of the shooting gallery and Ferris wheel that Mangels created have been found: click. If nothing else, you might be able to track down info on those "bullets" that were used, possibly even get leads on where to acquire a couple for your own collection.
Apparently, W.F. Mangels had a manufactory on W. 8th Street, between Sheepshead Bay Rd. and Surf Avenue. Uncertain what, if any, of it still exists, whether artifacts can still be found, whether artifacts where ever found in any clean-up over the years and where those might have gone.