Cleveland may have been the first family who infiltrated a union back in the 1920's through Tony & Frank Milano.
Unless I am overlooking or unaware of something, the earliest LCN involvement in a labor union, at least on a substantial level, is when Genovese captain Joseph "Socks" Lanza created Local 359 of the United Seafood Workers Union in 1923.
The top 3 families in the teamsters union historically was The Genovese through tony pro. Chicago through dorfman Lombardo and senese and the luchesses through Johnny dio and tony Ducks