For all my life, this question has plagued me. But after spending a few days in Lancaster County with the family, I now know that yes, they do. They don't use clubs; they use "bats."
Not only do they play golf, but apparently they have no qualms with using a golf cart either. I was told this is permissible because technically they do not own the cart; renting or borrowing or leasing such a contraption is perfectly acceptable under the rules.
Bear in mind I've done zero research on this. I'm just repeating what I was told by a fellow non-Amish guy who's lived in Lancaster County for several years after we spotted an Amish foursome on the course.