Epiphany Davis arrived at work in lower Manhattan on a recent morning, consulted her cellphone and set off by foot in search of products ordered via text message by wealthy New Yorkers.

From her company’s loft-like headquarters, Ms. Davis walked to a health food store to get SmartyPants Kids vitamins, but the variety was out of stock. Checking her cellphone often for instructions, she walked to a grocery store for a single bag of Guittard milk chocolate chips. She rode the subway to a Nespresso store for three boxes of coffee...