When I started Yale's student EA group in 2014, we tried a bit of this (albeit with pharmacies, not grocery stores). IIRC, we got as far as a meeting with CVS's head of corporate social responsibility (CSR), plus a few other conversations.

The companies we spoke to were choosing large, well-known charities. This was partly because of their branding (easier to pick up positive associations from charities people have actually heard of), partly because big charities tend to have highly appealing missions (e.g. St. Jude's, which has used its "free care for children with cancer" pitch to become America's fourth-largest charity), and partly (I'd guess) because the charities were easy to work with thanks to their size and staff capacity. 

I also suspect, from these and other CSR-related interactions I've had, that changing a charity choice is hard once it's been made. The professionals I meet tend to form relationships with the charities and staffers they work with, and it's hard to tell someone you've fired them for a more effective charity (forgive the link, it was too easy a joke to make).

I think this would be challenging, but might be worth pursuing, or at least trying, for the learning value. It involves the project of social change, changing attitudes and engaging the non-EA community, learning about their attitudes towards widening moral circles, the ways they are misinformed about the effectiveness of GH&D charities in LMICs (and maybe about farmed animals, etc.; although that could be a stretch) and whether this drives their attitudes or the other way around (see my project here on 'barriers to effective giving'.

This sort of project that could be attempted by students/student groups or part-time volunteers, and it might be motivating and fun.  E.g., I thought, "in UK university towns, where students often come from around the world, why do the local coops almost always only have a box for 'local charities'"

Ideally, they would do so in a coordinated to  organize and their efforts, collect and share their insights, etc.

Somewhat relevant to the EA Market Testing project, which is not currently very active. (However, Lucas Moore at GWWC is coordinating some efforts and collaboration and this might be relevant to him/them.)