Great place to eat and I have had no issues. Usually I go with a salad and grilled chicken. I do let them know I have an allergy.
They also have a gluten free bun that is sealed in plastic. They are not the best, but at least an option.
https://www.chick-fil-a.com/stories/inside-chick-fil-a/2017/06/19/whats-gluten-free-at-chick-fil-a
— Somewhat
— Gluten-free items are marked on the main menu
— Some
— Excellent