The photo claims to be from 1939, but I guess it could be from later. A quick research shows t-shirts were almost exclusively plain and white, with some color options for kids t-shirts until the 60s/70s. The only printing on them seemed to be for military t-shirts that would say what unit or part of training e.t.c.... the soldier wearing it was in. Also some workers on the movie "The Wizard of OZ" had some shirts that said "OZ" on them.
So this begs the question: was there actually a small market for graphic tees back then? Did Disney or other companies produce shirts with their characters or logos on them? Or is it more likely the above shirt, among others, were made by the people wearing them? In the research I found, it wasn't a regular practice to advertise/make graphic tees until about the 1970s.
It’s not easy to say when exactly the first graphic t-shirt originated, but an early example would have been from around the same time as this photo, which was 1939’s The Wizard of Oz, where the workers fixing up the Scarecrow are all wearing shirts that say “OZ” on the front in bold letters.
From what I could find regarding the image in question, however, was this:
The woman in question is named Sarah Elizabeth Turner Taylor, a name I only found through reverse image searching. She was only 24 when this photo was taken, and naturally there isn’t much information on her. Her memorial page, however, indicates that she had an interest in tanning hides, so I don’t find it to be a stretch that she would have made this Mickey tee.
Disney did have merchandise at this time, but it mostly consisted of dolls and comic books. The Walt Disney Family Museum states that:
”[Herman] Kamen wasted little time realizing his vision. Soon Mickey and Minnie could be found in department stores everywhere, adorning such products as: napkins, wallpaper, books, phonographs, all types of clothing, hairbrushes, toys, and much more. Mickey products extended beyond the store shelves as well, thanks to annual, then biennial, merchandise catalogs published by Kamen.
And from that - as well as the fact that Kamen signed on to make merchandise a mere seven years before this photo - we can conclude that clothing featuring the mouse was commercially available, but images of what the clothing may have looked like do not seem to exist online.
TLDR: The shirt may be a custom made piece of clothing considering similar interests held by the photo subject, but considering the merchandise put out by Disney at the time, it’s not conclusive.