Ashley Sandall recently set aside the keys to her Porsche sport-utility vehicle for three months to try out a few Cadillacs. When the Caddy she was driving proved too small for her weekend plans, she ordered up a different model with her smartphone. It would arrive the next day.

Ms. Sandall, a 35-year-old fashion executive who lives in New York City, used Cadillac’s brawny Escalade SUV for trips to the Hamptons and a football game in New Jersey. “It was great to have your pick of the vehicle best suited for what you’re doing,”...