The crash was reported just before 1 p.m. at the store located at 3475 Sonoma Blvd. in the Vallejo Plaza Shopping Center, Vallejo Fire Engineer Kevin Brown said.
The driver complained of chest pains and might have suffered a medical emergency, Brown said.
Ambulances responded and took four injured people to a hospital.
The two others declined transport. The injured were cut by glass from the shattered front window or were hit by the car, Brown said.
The car drove 20-25 feet into the store, which did not appear to sustain any structural damage, Brown said.