There is actually an app called BNB: Amazon Buy Now Button that should accomplish exactly what you are looking for. The app offers a 30-day free trial and then costs $4.99/month if you decide you would like to keep it.
I hope this helps! If there is anything else I can help you with, please let me know.
Kindly welcome to the Shopify Community.
To get your customer to be able to use their prime account your whole purchase process would have to be done on Amazon. You’d have to have each Shopify product page link directly to each Amazon product page, so your customer could log in on Amazon and add the products to their cart and complete the purchase on Amazon.
To properly add your button linked to Amazon products, you should get a meta field app, especially if you have lots of products.
Finally, you should add the HTML button, referencing the meta field in the product-template.liquid, or wherever your theme has the add to cart button.
Please to find product-template.liquid, Go to Online Store -> Theme -> Edit code then go to sections/product-template.liquid
Another way would be to try this Shopify app.
Please let me know if you have any further questions.