Police were contacted at 9.34am on Wednesday with reports of the collision.
The car then hit the wall of the pharmacy building.
Two women who are believed to have been in the building at the time are said to have sustained minor injuries - one with cuts and scrapes and one with an unconfirmed broken arm.
Police have been unable to confirm if the two women were staff or patients.
The car driver and motorcyclist have both also suffered injuries.
The hospital’s car park is closed to allow emergency services to attend and recover the vehicle and the pharmacy and laboratory building has been evacuated.
But the hospital remains open, staff confirmed.
Wendy Scott, chief operating officer at York and Scarborough Hospitals NHS Foundation Trust, said: ““Once we have an assessment from the structural engineers that the building is safe, staff will be able to return.
"In the meantime, an incident control centre has been set up and business continuity plans are being enacted to enable the hospital to continue running as normal, or for alternative arrangements to be put in place where needed.
"These plans are tried and tested, and enable us to continue to run services when there is an interruption to normal business.
“Our thanks go to everyone who acted quickly to evacuate the building, to help those who were injured in the accident, and to keep the scene safe and secure.
"Thank you also to those who are working hard to keep services running for our patients. Anyone who is due to attend the hospital for an appointment today should attend as normal.”