I feel like this is a problem the non-profits should tackle. The mother in the article does this, it seems she really is trying to work and not depend on charity/welfare. However, there is some financial literacy / access to capital / cashflow planning type issue going on that prevents her from planning properly (buying in bulk ahead of time). If the non-profit simply offered her affordable diapers (I imagine removing the retail markup alone saves a huge amount), a line of credit or something and helped her plan her diaper supply needs it seems like it could be a helpful service (maybe they do that, article only made it sound like they give away diapers to the poor)
Not everyone exhibits basic competency and planning ability, but that doesn't magically make other people responsible for setting up surrogates for them.