The incident happened on Monkmoor Road at about 4pm on Sunday.
It is believed the van had its blue lights on and was responding to an emergency when it swerved to avoid a car emerging from a driveway and mounted the front wall to a property.
Witnesses said the female driver of the car was taken to hospital.
A neighbour said: "She was pulling out and they just swerved and hit the wall.
"She's been put in the back of an ambulance. I think the police were alright."
Several concerned neighbours were outside observing as emergency services worked. The road was closed immediately after the crash happened. Traffic was passing in both directions by about 4.45pm, though the police van was still at the scene, along with fire crews that were making the area safe.
Another neighbour said: "It made a hell of a bang, I was just in my garden enjoying the sun and I heard this noise. I think everyone's alright but it was a shock, that's for sure."
One fire engine, an ambulance and several police vehicles were at the scene of the crash, just yards from the force's Shrewsbury HQ.