The nation's largest grocery chain stepped into the driverless delivery market Tuesday, bringing milk, eggs and other items to a customer's home in a vehicle with nobody at the wheel.

Although limited to delivering within about a mile (1.5 kilometers) of one Arizona supermarket owned by Kroger Co., it represents the latest step for industries trying to lower delivery costs of everyday items and those trying to launch self-driving cars on public roads.

Tuesday's delivery arrived at Shannon Baggett's house in the Phoenix suburb of Scottsdale. She was already receiving groceries weekly from large...