The pedestrian-friendly cities I know often allow vans and trucks to resupply stores on the walking streets, even if normal traffic is disallowed. They’re also encouraged to deliver in the morning.
Trying to point the issue to disabilities is often extremely counter-intuitive; it’s often hard for disabled people to use a car for everything (picture wheelchair transfers every time), as well as walking across huge parking lots or inside megastores. It’s often far better if they can just make it to a small store directly without excessive worry about high-traffic crosswalks. Public transit is often wheelchair accessible by default.
The mindset of completely banning cars is not one I’ve joined up with; as you say, contractors or the slim minority of workers transporting heavy goods should likely still be using cars. But that experience of driving is often terrible when every single person (on their own with no heavy cargo) is using a car for every trip.