The nanny may be willing to do the shopping, either with the children or outside of normal shift hours (paid for her time, of course). In that case, a prepaid card would work best, both parents contributing to the card and the nanny turning in receipts after each purchase. If she's the only preparing the food, she shouldn't need a list after the first week. If she's just reheating the food, but not cooking it, it's not appropriate to have her do the shopping.
If the nanny doesn't want to shop, then the parents should probably switch shopping duties, just picking up the required items while doing their normal shopping. 2 17 month old toddlers don't eat very much, so except for major trips after vacation or the initial stock up, it's likely to be about the same amount every week.