At least some of the people injured outside Main Street Elementary were children, though it was not immediately clear how many, city fire spokesman Erik Scott said.
The crash occurred at 2:50 p.m., shortly after school had let out for the day, according to a statement from the Los Angeles Unified School District.
The school is about five miles southwest of downtown Los Angeles.
Scott said it was unclear whether the driver is among the injured, under arrest, or being sought.
Helicopter footage from NBC4 showed a child in a pink T-shirt being loaded into an ambulance and a Hello Kitty backpack lying in the street nearby. The footage showed a silver sedan on a sidewalk and a black car on the street next to it, both draped in police tape.
ABC7 in Los Angeles reports the driver says he is 100 years old and thinks his breaks failed.