The lorry collided with the scaffolding by the Co-op in Market Square before 6.30am.
A spokesman for West Sussex Fire and Rescue Service said on social media: "Scaffolders are on scene making the structure safe, and our crew from @43Midhurst are liaising with the @RSPCA_official to ensure the welfare of the animals being transported in the vehicle. Please continue to avoid the area for the time being."
Midhurst's fire crew said the scaffolding has been 'safely removed' and is now en route to its destination.
The town centre remains closed at this time.