It happened at the junction of West Cromwell Road and Warwick Road in Earl's Court, west London, at 15:20 BST on Wednesday, the Met Police said.
The woman was taken to hospital after treatment from paramedics.
Buckingham Palace said the duchess's "heartfelt thoughts and prayers are with the injured lady and her family".
Inquiries into the circumstances of the accident are ongoing, the Met Police said.
A Buckingham Palace spokesperson said the duchess was "grateful for the swift response of emergency services and will keep abreast of developments".
"Further comment at this time would not be appropriate while the incident is being investigated," they added.
The injured woman, whose family has been informed, was treated at the scene by the London Ambulance Service (LAS) and London's Air Ambulance.
The LAS said she was transferred to a "major trauma centre" in the capital.
The Directorate of Professional Standards has been notified about the crash.
The Independent Office for Police Conduct (IOPC) has begun an investigation, which is at a "very early stage", a spokesperson said.
The watchdog had sent investigators to the scene on Wednesday night, they added.