Is there a way that you can use amazon music on the Apple Watch

At the moment, only if Amazon will allow you to physically download the music and transfer it to the Apple Watch so you can listen off-line.

Pandora's paid service allows this. I think there is a way (might be a hack) that allows Spotify to do this as well.

Whether Amazon provides an Apple Watch app that allows streaming Amazon music is something you would have to ask Amazon.