Shorts are permitted by the dress code. BTW I often find it too cold for shorts in some of the restaurants, so you may want to try for one pair of pants, just in case. FWIW.
He doesn't even have to wear a collared shirt (but you don't have to tell him that). Jeans, shorts, tees, sneakers are all okay at the signatures. No showing up in your swim/beach clothes. My DH is most comfortable in his khaki shorts and a polo or button down disney camp style shirt. He fits right in.
From what I have seen eating at many TS restaurants the last 3 years, including California Grill which is supposed to have a stricter dress code, you will be okay as long as you don't show up in swimming trunks and even then I would be surprised if they turned you away.