The lorry collided with the scaffolding by the Co-op in Market Square before 6.30am.
No injuries have been reported.
Scaffolders and Sussex Police had been working to make the area safe again and had support from fire crews in Midhurst.
As a result of the incident, traffic is slow and queuing in East Street and Angel Street and people are being asked to avoid the area.
"One crew from @43Midhurst is in attendance, to keep people safely away from the scene, while waiting for the scaffolding company to arrive."
A spokesman for Midhurst Station 43 said: "06:25 - W43P1 ordered to Market Square, #Petworth to a lorry in collision with scaffolding. No persons injured. Scaffolders and @SussexRoadsPol on scene formulating a plan to make safe. #oncall #Midhurst"