There is a small car rental company in my town that does just this although it's more like an hour rather than 15 minutes. They've got all of my details on file so it's as easy as a 10 second phone call because the person recognises me by voice.
I saw this service mentioned in another HN article about YC life last week and thought it was interesting: https://drivekyte.com/
A driver delivers the car rental to wherever you want then leaves on an electric scooter.
But there are no car rentals which would bring a car to your door in the next 15 minutes.