Is Dropshipping allowed by Amazon?

To dropship via Amazon, you need to follow Amazon policies and terms.

Please check the links below. 

1. Amazon Dropshipping Policy
2. Amazon Prime Terms 

Amazon Prime isn't available for customers (dropshippers) who purchase products for the purpose of resale or use Amazon Prime to ship products to their customers or potential customers. therefor you can't use prime products for dropshipping. 

To apply amazon dropshipping policy we recommend of using 3rd party warehouse services. some of the benefits using warehouse services are: control your inventory / handle returns being able to put your own branding on the box.

Here is a great shopify article about 3rd party warehouse services which cover most of the questions about it.