I've seen it done here, although only at the slightly "greener" store (Wild Oats). Not sure how the larger chain (Kroger, Publix) management would react. A cable lock is enough here; I just lock to the cart corral, bike rack, or some other metal railing.

As an aside, I've never had problems with the drive-up at the bank. I think few enough people are on bikes that there's no official policy.