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jeff sMarty Appel's book Slide, Kelly, Slide, suggests that Mike "King" Kelly was the first major figure whose autographs were pursued as they are today. That is, kids would chase after Mike trying to get him to sign a piece of paper or something.
Appel's book is not terribly reliable and is very poorly documented, but this is plausible, given Mike's superstar status and the fact that virtually all of the baseball player autographs from before 1890ish come from letters, contracts, and the like.