Two people were trapped under the Buick when it crashed through the glass front of the Tutoring Club about 12:30 p.m. PDT and came to rest against the back wall, The Orange County Register reported.
The car had been parked in front of the building before the accident, the newspaper said.
"We all of a sudden heard a large screeching noise and then just smashing glass and kids and people screaming," the center's owner, David Hill, told KCBS-TV, Los Angeles. "The car had gone through to the back of the entire storefront, there was rubble and just mess, everywhere."
KCBS-TV said the driver was 76-year-old Peter Hauk of Huntington Beach. Hauk received minor injuries.
One victim, identified as tutor Matt Martinez, 32, required surgery, while another, a 67-year-old woman, suffered moderate injuries, the TV station said.
Two 15-year-old boys suffered minor injuries. The sixth victim's injuries were not reported.