Soundiiz is using the Amazon Music API to access user music account. Due to this how this developer access has been made; you could have to do an update on your Amazon Music account to allow us accessing your playlists.
Go to your Amazon Music account and select one of your playlist. Edit this playlist to set it to "Public" / "Share" the playlist. A popup will ask you if you want to update your Amazon Music account privacy setting.
You have to confirm the operation. Once done, reload the Soundiiz interface and your playlists Amazon Music will be listed.