Very nice Bryan.
It looks like it may have been from 1911. A similar and possibly identical item (no back scan) was auctioned by Heritage as a postcard dated 1911:
https://sports.ha.com/itm/baseball/c...ription-071515
Here is a blurb from the Chicago Tribune on January 27, 1911.
As you probably know, Tinker had his hands in a number of offseason ventures over the years--seems he was always tinkering
- and his vaudeville act was performed prior to 1911 as well. A couple of other articles I saw seem to indicate that 1911 was going to be his last hurrah on the stage, and that his circuit of appearances was going to end in the city of Chicago itself. It therefore is possible that your item is from the end of his run, although I suppose it could be that the same type of postcard was re-used from year to year too.
UPDATE: It seems Tinker was largely responsible for helping to create the "West Side Rooter's Association" in January 1908, which was formed to provide winter entertainment to the locals. It does not appear anything specific was created in the month before Joe headed to Spring Training, so 1909 would seem to be the earliest for this postcard or whatever you wish to call it. Because Tinker's vaudeville act from 1909 was a skit called "A Great Catch", and was billed as such, and the 1910 performance was called "A Home Run", it's likely that "Oh You Short Stop Kid" was a separate production and probably from 1911.