I'd be wary of fresh fruit and veg via an online supermarket but we can get that from the local grocers.
The pickers (in Tesco at least) just pick from the front of the shelves so if they are doing the shelf stacking right it will be stuff closest to it's date. But most dry stuff that is 6 9 or 12 months into the future so hardly relevant. For fresh/perishhables I might look at the date and then dig to the back of the shelve but not very often. But then I only use the various dates as a guide, look and smell are more influential on deciding if something is edible.