The collision, which also involved a blue Honda Accord and a white Toyota Camry, was reported shortly before 11:20 a.m. at 22nd Street and Potrero Avenue.
Passengers at the scene said the shuttle had been rear-ended.
When firefighters arrived, they reported between 15 and 25 potential patients, but only five required transport to the hospital, the dispatcher said.
A UCSF employee who was on the shuttle when the crash happened said the shuttle was stopped at a light when she heard a crash and felt a bump.
"We hit our heads on the headrest of the shuttle seats because we don't have seat belts in the shuttle," said the employee, who declined to give her name.
She said two of her coworkers and the shuttle driver were among those taken to the hospital, but that their injuries did not appear to be serious.
At about 12:30 p.m., Potrero Avenue was still shut down between 21st and 22nd streets, and all three vehicles involved were still at the scene. Deployed airbags could be seen inside the two sedans.