Children's Council provides free diaper supply monthly to families utilizing CalWORKs in San Francisco. If you have an Electronic Benefits Transfer (EBT) card, you can bring your card to one of 15 locations and you'll receive six packs of diapers each month per child under the age of three. Also, try Googling "Diaper Bank" plus your city name - there are many programs around the country and they usually use the phrase diaper bank to describe their offering.
One of my first projects when our baby was small was a natural homemade baby powder. His tiny little delicate rolls were difficult to keep dry with so many diaper changes during the day. I had so much fun concocting the perfect gentle, soothing baby powder that could be used on him, and it smelled so wonderful I ended up adding it to my own post-shower routine!
There are several different organizations around the country that distribute free diapers to needy and low income families. Many of these are charities or churches, with some government programs also assisting. There are programs for single mothers, teenage moms, and families living in poverty. Anyone that needs free or low cost diapers near where they live, and that meets qualifications, may apply.