I was searching through the online archives of the New York Times and happened to come across an article outlining all the details of the 1919 World Series, including the list of players eligible to play for each club.
You can find the article in the
Oct. 1, 1919 issue.
As I recall, the newpaper published similar pages for other World Series around that period.
EDIT: Just found the same kind of detailed article for the 1920 World Series in the
Oct. 3, 1920 issue.