Patrick, a Shopify Guru here :).
In our app store, there are a few apps that should help you out with this. For example, Stitch Labs and Order Desk should both be able to connect directly to your Amazon FBA account.
To see some of the other options have a look here.
Hope that helps :)
We have your solution! We've just released an app that integrates and simplifies FBA shipping and Shopify in the following countries:
The app is super flexible and pulls in all your Amazon FBA shipping & fulfillment fees. Which you can pass along to your customers, discount them, or simply provide free shipping. We include all 3 of the Amazon shipping types as well.
We are in BETA right now so its free, so come check us out!
Let us know if you have any questions
I am developer and I created this for a client.
This was a private app and public app development is currently in progress.If someone wants it I can setup it on his store.The payement is not recurring, its one time buy with setup.Please contact me on firstname.lastname@example.org or skype - manish.gautam97
If you are still looking for a solution, look at Amazon Auto Multi channel fullfillment , this app is published on Amazon Marketplace Appstore and Shopify.
Works for all Amazon marketplaces,
Let me know if this helps?