Young people are obsessed with the private equity-backed chain, which has been doubling down on sugary green tea.

Confusion ensued. Customers at Ms. Jaffe’s shop tried to cash in rewards in the Blank Street app, she said, while others emailed her catering requests for Blank Street’s mint green coffee trucks.

One of Blank Street’s mobile carts. The fast-growing coffee start-up will open its 12th location this week. Photo: Lanna Apisukh One of Blank Street’s mobile carts. The fast-growing coffee ...