assuming the question comes from the US - perhaps similar conditions apply in larger urban conurbations - but a national service might prove tricky.
Ocado is also worth looking at. Set up by 3 investment bankers, it sources its produce from one of the smaller, premium supermarkets (Waitrose) but has a (rather funky) purpose built distribution centre from which deliveries are made - rather than having a fleet of picker/drivers at retail stores across the country. Of course, this also limits their geographic coverage somewhat.
The biggest problem I've seen is that people don't trust someone else to shop for fresh things for them. Vegetables, fruit and especially meat have this problem. Tins and boxes of dry food are easier to trust.