Didn't come from a deprived area myself, but we had a teacher at our (admittedly small) school who would leave a selection of sanitary products, wet wipes and multi-pack supermarket underwear in pretty baskets in the school toilets, for girls who'd been caught out.
If anyone had been suffering period poverty, they could have emptied the basket in the cubicle without anyone knowing I guess, I doubt anyone would have cared.
Potentially too expensive on a larger scale, though reassuringly girls did seem to use them as needed and weren't just wasting them all and sticking them to the ceiling or something equally ridiculous.
I would say, that as so many young girls are shy around the subject, blanket provision rather than trying to target those who may need it, is likely to be the best approach.
As someone in marketing (the devils profession, I know...), I would contact all the big sanpro brands, plus independent companies like Pink Box and see if they would give you a sizeable discount on a bigger order. I'd even try those with less conventional products like mooncup and that company that does the fancy absorbant pants.
Or ask whether they'd be willing to sponsor your school in exchange for some good publicity, i.e. offer to contact the local press/daily fail and tell them about your new "partnership".
Companies can be incredibly generous if you know how to ask!
If you manage to negotiate a good discount, try approaching local "women's" groups to find funding. Would the local WI/guides/uni feminist society etc be willing to do a fundraiser? Or the PTA? They do like local charities where 100% of monies raised are buying useful products rather than paying an executives wage!
Just make sure if you provide tampons to include/display clearly (framed on a cubicle wall?) the safety leaflets - its probably more practical to just offer pads so the school doesn't end up being sued for its well meaning endeavours by a child that's developed TSS.
Sorry, I know teachers have no time as it is, and woefully inadequate wages, so my suggestions might be asking a bit much of you time wise.
PM me if you want, I will happily spend some time on this. I could put together a bit of an "action plan" and draft some letters/proposals for companies for you as this months good deed 