I like the idea.
There are some things, though, especially perishables, that aren't amenable to e-commerce.
I’m not sure what perishables Sears sells, but that’s a low-profit/high-waste market anyway. Maybe one they should leave behind.
Sears has some cred left in products like appliances, tools, and some furnishings. I would work to become THE go-to source for those items in the e-market, putting the capital saved from moving away from brick-and-mortar into building the best-in-class lines in those areas.
If you insist on keeping the K(rap)-Mart overhead, maybe that would be the place to locate your perishables.