Hopefully it's ok for a partial answer since I accomplished a little legwork. You ask about the authenticity and then about the history of printed sweatshirts (and though you don't mention it there are other notable things like elastic cuffs that make it look very modern).

I can help you with the first part of your question, this appears to be an authentic Dorothea Lange photograph from 1938, part of a series on Shafter migrant camp: https://loc.getarchive.net/search?q=Migrant%20child%20in%20Shafter%20camp%2C%20Farm%20Security%20Administration.%20California

Edit:

I did find something in the inexhaustible archives of this sub, from u/Chengweiyingji that actually asks about a very similar situation to yours (i.e. a late 30s printed/graphic piece of clothing)

https://www.reddit.com/r/AskHistorians/comments/jufr8d/graphic_tshirts_gained_popularity_in_the_60s70s/?utm_medium=android_app&utm_source=share

I'm not sure if the sweater in question is specifically Disney merchandise (edit2: I think this shirt is almost certainly a Disney product) but if so that lines up, but this might have been quite an unusual sight. It's also very possibly homemade, and some comments in the answer briefly discuss techniques.