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MWNot that anyone wants to know, but the first baseball cards issued with "gum" were from the 1888 E223 Series. A decade later, Whitehead & Hoag produced a series of baseball pins that were distributed with Cameo Pepsin Gum. John Colgan was actually one of the first to "mass produce" flavored chicle or gum, but did not include it with baseball ephemera until 1909.
Technically though, modern bubble gum was not invented until 1928 (by Walter Diemer, Fleer). In that sense, the Goudey Gum Company may very well have been the first to distribute "bubble gum" with baseball cards.