OP, Congratulations on what sounds like a great project!
I've done a lot of work in this area, and my impression is, when people really need food, they don't worry too much about who knows it. They just make sure they find a way to get it. So anonymity might not be as much of a concern for people as you think. However, I will also add the caveat that I work in the inner city, where there is far less stigma attached to being poor or hungry. And, I don't work with teenagers. If your school is in a suburban area, you might find that students at the school are more concerned about their image than almost anything else -- so very few students will admit to needing food.
I'd suggest that you have four options:
1) If you want to distribute food entirely anonymously, have people submit an anonymous form saying how many people they need to feed, and what food needs they have. You can assign each form a number, or have them choose a code word, or find some other way to designate each family. Then you can go through the food bank and make up "grocery bags" for each family, then label and leave them somewhere for those folks to pick up on their own, unwatched. Problems with this method: you might pick out stuff they don't need or want, or can't eat; people might take food meant for others; folks might still see people picking up their packages wherever they're left for pickup.
2) Forget about anonymity, and just publicize it as "if you need food, come and pick out what you need." That way you know that people are getting the stuff they'll actually use, but they also have to swallow their pride (in some cases) to get it.
3) Publicize a need for volunteers, in addition to calling on "needy" students. Advertise heavily with some phrase like, "come and help with our food bank -- and take food to people you know who need it." Encourage people to come and make up "food bags" for people that they know in the community who are hungry. That way, people who need it for their own households can make up bags and take them home, while pretending that they're for other people. The problem with this method: you might have a lot of well-meaning students dropping off unwanted or un-needed food for a whole lot of people, so your good intentions might be diluted by others' fervor. It's probably the best way to help needy students get what they need, though.
One important thing, I think, to mention: do NOT, by any means, make a big deal out of "deciding" who should get the food. Just make it available; let others decide what their own level of need is. There's no such thing as someone who is "more deserving" or "less deserving" of free food (or of particular amounts of food); everyone can decide for themselves whether or not they need/want the service you are offering. Preserve their dignity by respecting their judgment, no matter what you privately think of their real level of 'need.'
4) Expand beyond your school's boundaries and hook into a larger network. Choose a particular food center. soup kitchen, or food bank in your larger community, and direct your school's donations to it. They will be grateful for the added source of donations, especially if you can figure out a way to make the donation ongoing, consistent and sustainable (i.e., it will keep going on a regular basis after your project is over). Let them figure out who the hungry people are (hungry folks have already found them, anyway) and get the food to them. You just make your project about getting more donations to the food banks. The positive things about this plan -- you get linked into the whole community of people who are trying to feed the hungry... and you can learn from them.
Good luck with your project!!