The 7-Eleven next-door hadn't opened yet when Kyung Yu started to worry. The owner of Kyung's Fruit Store in Chelsea for more than 25 years, Mr. Yu even contemplated sending the company a plea for mercy this summer, warning that its arrival could spell doom for his business.

About a week ago, the convenience store's signature green and red sign was up, and the store was open. Mr. Yu is bracing for the worst. "If my business goes down, I don't know what I'm going to do," said Mr. Yu, whose offerings include Asian candy, orchids,...