Amazon Music for Artists is available on both web and mobile.
Click here for web. On mobile, head to the Apple App or Google Play Store and download the Amazon Music for Artists app. Once the app is open, you will be directed to sign-in using an Amazon retail account. If you do not have an Amazon retail account, you will need to create one in order to continue going through the sign-up and artist claim flow.
To claim an artist:
Search and select any artist.
Press "Claim This Artist".
Fill out the requested application fields. They also provide the option for anyone to connect with their artist's social media accounts for faster verification. The more information you are able to provide, the easier it will be to verify your relationship to the artist you are claiming.