I have access to my employers online grocery account, and they let me add whatever I'd like to it each week. In the event that I forget to do so, MB will either shoot me a reminder text over the weekend, or add what she knows I usually get/like.
Admittedly I felt awkward about this at first, seeing how much my stuff would add up to, but I'm more or less comfortable with this arrangement now, and grateful to my employers for their generosity in this regard.
Also, since I'm a vegetarian and order mostly different items than what the family eats, I never run into the problem of accidently polishing off something that they may have wanted.
As PPs mentioned, every family is different in what they can afford and are willing to offer their nanny in regards to lunch.
Since your nanny's an adult, and you've already put your offer out there, it's best to just leave it up to her at this point.
Perhaps you can occasionally remind her that she's more than welcome to help herself to certain foodstuffs in the fridge and/or pantry. Either she'll get comfortable doing so, or she'll continue to do what she's already doing.