Debbie Masters, who was 37, died in hospital after the crash in Battle, east Sussex, on Sunday.
A family statement said she was "a loving wife and mother, who was adored by her husband and will be missed forever".
Her children suffered shock and minor injuries and their father was taken to hospital with a serious head injury.
Mrs Masters had been walking along Upper Lake in the town with her family - one of them a baby in a pram - when the VW Beetle hit them before it crashed into a cafe.
Mrs Masters, who lived in Battle, suffered critical injuries and later died.
Customers were trapped inside the cafe, but none of those inside were injured.
Officers have said they are investigating reports the 34-year-old driver, who had shock and minor injuries, may have been under the influence of alcohol at the time.
Sgt Peter Verney said blood samples were taken and sent for analysis.
He said: "Our thoughts are very much with the family at what is a very difficult time."
Police still want to speak to anyone who saw the collision at about 13:55 GMT.
Tributes have been paid to a mother of three who died after a car ploughed into her as she walked with her family.