Firefox doesn't have a built-in feature for this, but there may well be an add-on that can do it. It doesn't even need to be specific to Microsoft's mail site, it might be something that looks for dynamic changes in a webpage.

You could start by searching on the add-ons site: https://addons.mozilla.org/firefox/

Kev1nt said

Hello I have Firefox ver 1.32 and i use the browser to go to my hotmail account. I need to set up some sort of audible alert for when I recieve New emails Can anyone help ?? I contacted microsoft and the person I had a live chat with addmitted she was new inexperienced and couldnt help neither could the person she asked to help. There must be someway of connecting a windows sound with a new incoming email, at least I hope there is. Thank You Kevin

Sorted Thank you very much X-Notifier 3.5.6 does the job