A 2005 Toyota Avalon went 50 feet inside the store, located at Redwood Road and Grove Way, at about 3:45 p.m.
"It went pretty far (inside)," said Alameda County sheriff's spokesperson Lt. Ray Kelly.
Authorities do not believe the driver intentionally drove into the store nor do they think drugs or alcohol played a role, CHP Officer Dan Jacowitz said.
The driver was an elderly person who was in the parking lot and accelerated into the front of the store where cashiers were tending to customers, Kelly and Jacowitz said.
Four to six people were hit by either the car, debris or other objects. All of the injuries were minor or moderate, Jacowitz said.
The driver is cooperating with authorities, Jacowitz said.
There are no further details at this time.