WYOMING, Mich.—Like many adults with disabilities, Nathan Mort has often struggled to find and hold a job. A conservation group once declined his attempt to volunteer. The 37-year-old West Michigan native, who has a high-functioning form of autism, ended up living with his parents and dependent on government payments.

His fortunes turned several years ago when a local food distributor, Gordon Food Service, found itself short of entry-level workers and developed an internship program for adults with disabilities. Mr. Mort...