So, yeah, in theory you can get your groceries delivered to you, but none of those services are intended for cash-strapped families and/or conscientious shoppers.
The prospect of lugging 4 bags of groceries plus 3 kids on a late, crowded MUNI bus is just sooooo inviting...
I realize this is a bit of a chicken and egg problem but we have seen how quickly startups or even technology being implemented by existing players can respond to increased demand.
However it's also not that common for people here to carry lots of groceries at a time, unless you're planning a party. It's a lot more common to just pick up a few things each day: instead of one "big" shopping trip a week, most people make a short 10-minute stop every day or two, often on the way home from work. Partly for convenience, but partly also because that way your food is fresher. If you're shopping in that style, your groceries will fit in a regular bike's cargo basket.
Other supermarkets do this a lot cheaper. Or free.