SHENZHEN, China—In a warren of tiny shops beneath grimy residential towers, a white-haired man selling Snickers bars and fizzy drinks from a kiosk no larger than a cashier’s booth is figuring out a way to move $100,000 out of China.

That is twice what Chinese are allowed to send out of the country in a year. Licensed banks won’t do it. But middlemen like Mr. Chen, perched in his mini-mart at the front lines of a vast underground currency-exchange and offshore-remittance network, can and often will.

“There’s...