I've ordered the natural/grass-fed (not sure if it was locally source, probably not since it is encased in plastic, making it last longer) beef from Amazon Fresh and I'm pretty happy with it (I'm probably not thinking sustainably, sorry) because it doesn't spoil as quickly as the beef I buy from my butcher counter. Side effect is that it is usually less, by weight, than the stuff I buy in person, so I end up eating less, which is just fine. (Steak is, for me, hard to portion into leftovers, it is a mind thing).
I'm already buying everything I need from Amazon other than food... and since Amazon Fresh I've been buying food once per month, sometimes twice per month, as a Friday morning "no shopping trip for this weekend" move - in fact I spend less than I would have at Harris Teeter based on past experience.
I would have done this with Peapod etc but their website was lacking in 1) choices (Amazon will get you pretty much any food item) and 2) delivery window and 3) unattended delivery