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.
I will bet against Walmart. They have a high cost structure for their existing stores which makes them vulnerable to hard discounters but they also invest a lot of money into their online ventures to fight Amazon. It's a battle on two fronts and will rip the company apart.
In the past, Walmart failed hard in Europe but ALDI and Lidl even conquered UK and Australia. As you might now, Europe is far away from a single culture and yet both chains were able to win in every country due to adoption and experience. While they are no franchise concept, there strategy reminds me of McDonald's.
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.