Thanks to Go Dairy Free viewer Christine for emailing this recommendation:
"In Downtown Disney, California Ralph Brennan's Jazz Kitchen Express is very dairy-free friendly. The breading on their food is dairy-free, the fries are dairy-free, even the beignetes are dairy-free. Their special mayo is dairy-free too. We had a lovely meal there and I thoroughly enjoyed myself."
Location: 1590 South Disneyland Drive, Downtown Disney® District, Anaheim, CA