I'm not sure, but they do have a "sharing table" in the lunchroom, where kids can leave unopened snacks or other foods for children who need something to eat at lunchtime.
Are you involved (or, if not involved, connected) with Scouting organizations? The assembly of food backpacks sounds like something Scouts aged 10 and up (roughly Webelos in Cub Scouts and Juniors in Girl Scouts) could take some time, especially if you have Girl Scouts earning their highest awards (Bronze, Silver, Gold) and need community service hours.
I'd also make an effort to connect with youth groups at your local houses of worship - my Youth Group would have loved the way to do a day of service (or multiple days of service) in this way to help Confirmation candidates earn their community service hours, among other things.
The local food bank runs an annual "
Springtime Spruce-Up" collections program for personal care items, like (new, unopened) toothbrushes, toothpaste, bar soap, shampoo/conditioner, lotion, shower gel, etc. ... and tampons/pads ...
Some of the items are donated by local businesses; others are collected in a community-wide drive.
They are assembled into care packs by volunteers working at a local church on an evening sometime in May.
It's a huge undertaking, but people in our community, from Scout troops to service organizations, are happy to pitch in. My Scouts helped assemble the care packs about 2 years ago; they were 8 and 9 years old at the time and they had all kinds of little jobs to pitch in.