There are two grocery delivery services.
One is Garden Grocer and they have their list of grocery items and prices posted on their website. You order from them, at their prices, and they make the purchase and deliver to you. I think there is some markup in each item.
The other service is WeGoShop. They will shop anywhere you want (including discount places) and pay the prevailing price in that store. They bring your receipt, add a small charge to the total, and that's what you pay.
I don't know how the overall cost compares, but because of their business model WeGoShop obviously offers more flexibility in store, brand selection, and price.
Both services had bumpy patches in their service several years ago, but both have gotten very good reviews for several years.