Seattle area residents can contact WestSide Baby for free diapers. The non-profit organization is mostly run by volunteers from the community. They currently distribute almost 300,000 diapers to the low income, unemployed, and working poor in Washington. They also partner with numerous local organizations, including more than 100 social service agencies and charities. People can not only get free diapers from the organization, but they can also get toys, clothing, and baby equipment like cribs and car seats. Call (206) 767-1662
With Pampers Rewards, you need to download an app to scan in the in-pack codes. I did it and it's actually super easy. You get 100 points for signing up, 50 more when you scan in your first code. The one hitch is that while Pampers Rewards has loads of stuff you can get with relatively few points, it actually takes a lot of freaking points to earn a pack of free diapers. I did the math, and it would take 16 purchases of 166-count Pamper Swaddlers Size 4s to earn the 2100 points required to get a free pack. Now that's a lot, but also, you're going to buy the 16 packs anyway, so you may as well get one for free.
Sometimes, a written letter or email telling a diaper company about your baby’s birth (especially if you have multiples) asking for a product to sample can result in some free diapers. I wouldn’t really recommend going the “sob story” route too strong, but companies want to know you, their customer. If you ever have faulty diapers, be sure to contact them as most have a guarantee and will replace your defective diapers for free.
There are resources available to you if you need them, though. Also, don't discount cloth diapers if you have the means to launder them. They're often easier to get at free or reduced cost, not to mention they're way better for the environment, and you can be super smug about that. (Unrelated, but once you're done diapering your baby, those diapers really make the best dusting cloths and glass cleaners.)