The Amish do fly, quite often.
Used to often see them coming and going through BWI.
Since the issue has been raised in this thread I think the Amish are a wonderful example of how people with strong views on how life should be lived should live.
They choose to live in their communities the way they want to, they don't try and exclude others from their communities and they don't cause trouble. They don't demand that the host country change their laws to accommodate them and their culture.
Mexicans and Muslims take note.