You can use a screen recorder or audio capture tool while the app is ringing to save a local copy of the audio. How to Manage Your Amazon Chime Ringtone Settings
Look for the checkbox. Here, you can enable or disable the default chime for incoming messages and calls. On Mobile (Android & iOS) Changing mobile app settings - Amazon Chime amazon chime ringtone download
If you want to use this specific sound for other purposes, you may need to: You can use a screen recorder or audio