A Shopify account (you can get started with a free 14-day trial)
Professional Amazon seller account ($39.99 per month + additional selling fees)
GS1 Product bar codes
The bar codes are particularly important since Amazon will ban seller accounts that don't have them (UPC bar codes for US and EAN bar codes for Europe). You can use Shopify's free bar code generator.
To sell on Amazon using your Shopify store, you need to add it as a sales channel:
Navigate to the left menu of your Shopify admin and look for the section that says "Sales Channels", then click on the + button beside it.
Scroll through the next window and look for Amazon, then click on the Add button beside it.
Confirm the process and link your Shopify store with Amazon.
Note that the Amazon sales channel is only currently available to US sellers.
Also, your listings on Shopify don't automatically show up on Amazon after adding it as a sales channel. You will have to create the listings on your Amazon seller account and link the checkout process to your Shopify store.
Requesting Fulfilment for Orders on Amazon
Once you receive an order from a buyer on Amazon, you will receive a notification to review and request fulfillment for said order by doing the following:
Click on "orders" on the left menu of your Shopify admin.
Find and click on the order that you want to fulfill.
Review the order details, and if you're happy with it, click on "Request fulfillment".
Click on "Send Fulfilment Request". Your credit card on file will be charged the cost of the product and the vendor ships the product directly to the buyer on Amazon (up to 14 days lead time).