I've grocery shopped exclusively with my eBig Dummy for the past few years. Initially my regular store's manager objected, but when I politely but persistently pointed out that I'd been a weekly customer over 10 years, that the store lacks any secure rack, that bike theft is rampant in the vicinity, and that my BD actually is my shopping cart, not (just) a bicycle, she eventually relented but only with the (acceptable) proviso that I always have a hand on it so as to prevent anyone from bumping it and having it fall on them.
While I wouldn't say I've always felt welcome thereafter, I've managed to avoid further hassle by shopping early when few other shoppers are about. I've also shopped at more distant grocerys at off-peak times with the BD without complaint and would travel farther and change my patronage should my BD ever be banned.
Interestingly, like others above, the one time I tried to shop at whole foods, I was also required to lockup outside. At least there was a rack, and the store is in a less dodgy area, nevertheless, they've lost my business.