Since 2006, when the insurgency in Afghanistan sharply intensified, the Afghan government has been dependent on American logistics and military support in the war against al-Qaeda and the Taliban. But to arm the Afghan forces that it hopes will lead this fight, the U.S. military has relied since early last year on a fledgling company led by a 22-year-old man, Efraim Diveroli, whose vice president was a licensed masseur. With the award last January of a federal contract worth as much as nearly $300 million, the company, AEY Inc., which operates out of an unmarked office in Miami Beach, Fla.,...