Faced with a prostate-cancer diagnosis more than a decade ago, Ben Carson, the Republican presidential hopeful and retired surgeon, consulted an unusual source: the medical director of a Texas company that sells nutritional supplements made of substances such as larch-tree bark and aloe vera extract.

The company doctor “prescribed a regimen” of supplements, Mr. Carson told its sales associates in a 2004 speech. “Within about three weeks my symptoms went away, and I was really quite amazed,” he said to loud applause, according...