Yes, I actually managed to do this myself. You need to manually install the plugin here to enable audio drivers on your unraid machine: https://github.com/ich777/unraid-sound-driver
Then within the /home/default/init.d directory I added a new script called audioPermissions.sh:
getent group audio2 || groupadd -g 17 audio2 && usermod -a -G audio2 default
This should fix the permissions issue where there's a mismatch with the audio group.
Then restart your steam-headless container and it should have audio on startup.