Configuring preferences and options in Outlook.com
Dear Lex Lascuna
Thank you for posting to Microsoft Community
According to your description, I understand that one of your emails is not able to send out a notification sound to remind you of the arrival of your emails in the web version, if there is any misunderstanding in my understanding, please point it out.
You can check if you have turned on sound alerts for this account, you can click on the gear shaped [Settings] button at the top right corner of the outlook web version. After that click on [General] - [Notifications], click on the mail category, which contains a [Play Sound] you need to make sure that this button stays turned on.
This reply may not directly address your issue, so feel free to let me know how it goes any time by repling below.
I look forward to hearing from you!
Carmid.L-MSFT | Microsoft Community Support Specialist