We live in an age were BJ's and Costco are no longer the only game in town to get bulk supplies. Many websites are offering very competitive prices to the giant warehouse wholesale clubs. (Thank goodness, those places stress me out). We knew when we had Charlie that joining one of these clubs was probably going to happen.
For a while now, I've been seeing subway ads for sites that are going head to head with BJ's and Costco, but with free shipping and without the yearly membership fee. We've been using Amazon Prime forever and have always loved it. Very rarely has a package not shown up on time or damaged. We get most of our household items from their Subscribe & Save program. On some months this lands us 15% off Amazon's already low prices.
I had high hopes for either Boxed.com who really tout their competitive prices and free shipping over $50. I have a few friends who swore by Diapers.com for their kids. Jet.com was a new player, and technically did have the lowest overall price. But, they have a secret formula that they apply to your cart which lowers the price. I really didn't like not knowing how that worked of if it would change month to month.
In the end, Amazon won out. We get our usual 15% off Subscribe & Save the months we receive 5+ items (you play with it and game it a little) Diapers always get 20% off. If you find yourself running low, just hit the "I need it now" button and you will get your next shipment early. Bonus points: We also have an Amazon Credit Card (that is only used at Amazon) for an additional 1-3% cash back. We let this charge up and then every few years buy something nice with it. It's how we got our Sonos Playbar.
If you are interested in the spreadsheet above, it can be found here