The major player currently is Walmart for most Americans. It's investing into online at a rapid pace, they recently added free online pickup for groceries and purchased Jet and Bonobos, so I think the real war will be between Walmart and Amazon/WF.

Value basics and speciality stores like Aldi and Trader Joes (which have very spotty coverage currently in the US) and regional full-inventory stores (Piggly Wiggly, Harris Teeter, Bi-lo etc) will pick up the crumbs.