This Chrome Extension adds a notification sound when receiving new messages on Omegle. You may disable the sounds by clicking on the extension while on the website.
I did all the steps and it's showing up like an extension but no sound actually happens. Made sure I had it turned up, still nothing. I was using Edge extensions so I literally downloaded Chrome just to see if it only didn't work on Edge, but it didn't work on Chrome either.
Edit: Just wanted to add that it very randomly worked for maybe an hour and then turned off again. Got no clue what caused that, it was all in the same session. But it was very nice while it lasted!