Amazon has released a firmware update for the new Kindle Kids Edition that shipped in October 2019.
The new firmware version is 126.96.36.199, and it comes only a few weeks after the last update (0.5). There's no official description on the Kindle support pages, but I am told that the update adds support for Audible audiobooks. A user will be able to access the audiobooks available via their Kindle Freetime Unlimited account, and the account owner/manager (aka parent) can add Audible audiobooks to the Kid's profile.
This is the same hardware as the basic Kindle, so if you want to listen to audiobooks you will need to pair the Kindle Kids Edition with BT speakers or headphones.
But before you can do that, you will need to wait for the firmware to be installed. There's no option to download the firmware update and manually install it, for obvious reasons, so you'll have to turn on Wifi and wait for Amazon to push the update out.