Diapers cost at least several $100s of dollars annually per child. There are programs that provide them to families that meet certain financial guidelines.
Here's a national program that provides disposable diapers:
Here is their income guidelines chart:http://communitywide.org/diapers-for-babies.html
There is also a national cloth diaper program.http://www.clothdiaperfoundation.org/
This one sometimes closes applications because of so many applications, they have to rebuild their diaper supply.
Check in in often or signing up for their online newsletter will reveal when their diaper applications are reopen. Their Join Our Mailing List is about halfway down the page on the right.