1. The cost of MS license in the third world is really high comparatively, even with academic and non-profit discounts. I've worked at a school in South Africa where the licensing costs of a small lab could have hired an extra teacher. You don't get academic/non-profit discounts if you buy once-off licenses, to get those you've had to go with subscription model since at least 2003
2. The support offered by Microsoft at the school did not cover desktop support, or if it did, that support was so slow in coming as to be useless. That's one reason I had a job. IN this case using off-the-shelf software carries no support advantage, and little usability advantage. Chances are nobody at the school can use a Access, or even excel particularly well. From the original poster suggestion of running it on a VPS I assume they have internet, and are probably more familiar with a browser and word than excel.
All that said, the idea of just using a spreadsheet makes a lot of sense. From a reporting perspective, it makes life easy, and getting data to UNICEF et al is as easy as emailing a copy. I'd avoid Microsoft though, and go with libreoffice.