As you say, pensionpat, just asking basic questions about the households of food bank users should enable you to add sanitary products to the supplies of those with girls and women of the target age group. As for the wider community, the primary school I used to work in just left a basket of products in each of the appropriate toilets for older pupils or staff to take if needed. It wasn't a huge expense, and didn't take much effort. The cleaners just topped them up along with the soap, paper towels, etc. I'm really surprised that your local schools don't want to work with you, especially if you're offering to donate the products.
Perhaps it's time sanitary products were available on the NHS. I doubt that women who can afford them would take advantage of this - even though free condoms are available, most people surely still buy their own - but they'd be there for those who needed them.
The comments on here targeted at other posters and doubting or downplaying their personal experiences are completely uncalled for. Rude, patronising and unpleasant, as Roses says.