The BMW mounted the pavement and broke through the store front of Hackney Pirates charity in Dalston's Kingsland High Street.
The charity's education centre and shop was open at the time but no injuries have been reported.
Witnesses said the crash, at about 12:40 GMT on Sunday, sparked "mayhem" in the area.
"It was a busy day in the run-up to Christmas" when the car hit, Catriona Maclay from the charity said.
"We're extremely grateful everyone is OK."
A Met Police spokesman said the driver of the car was arrested on suspicion of dangerous driving and road closures remained in place into the afternoon.
Hackney Pirates runs educational activities to "develop the literacy confidence and perseverance of young people in Hackney".
The building in Dalston includes meeting spaces, coffee facilities and a book shop.
Ms Maclay said the damage would take days to fix, but the store was hoping to re-open before Christmas.
"The front of the shop doesn't look like the front of a shop any more" she said.
"We won't leave it shipwrecked because the windows are broken and we need to make sure it's safe for kids."
Twitter user @PaulieTandoori said the crash had led to "traffic mayhem" in Dalston.
A man has been arrested after a car crashed into a charity's building in a busy high street in north-east London.