Not sure how it works with other insurers, but we have a few policyholders who hold ''normal'' car insurance with us, but then have a seperate policy that covers them solely for the purpose of pizza delivery, etc.
We do Business Class cover - this allows you to use the vehicle for travelling for the purpose of work (ie if you're and area manager and travelling between stores, etc).
Delivery of goods would need to be covered under some sort of commercial cover that could run either alongside your existing policy, or as an alternative, but giving you full cover for all insurance purposes.
Try Gladiator as they are a commercial broker - they should be able to inform you a little better