Order Automator does this. It works out of the box for US and Canada marketplaces, automatically fulfilling Shopify orders via your Amazon FBA inventory.
We can also customize it to work with EU as well if you want, just give us a shout at email@example.com.
Our FBA Shipping app may be able to help! You can check it out here ----> https://bytestand.com/fba-shipping/ FBA Shipping connects your US & non-US FBA account with your Shopify store, allowing you to finally leverage the power of Amazon FBA around the world. Once setup your orders can automagically be sent to and fulfilled by Amazon FBA.