Are there any email sites that will make a sound when you get a new email if you leave the site open?
I don’t know of an email client that doesn’t, or of a web-based client that does.
The problem is turning the sound off, not turning it on.
Not that I know of, but that could probably be done with a browser extension. What operating system are you using, and what browser? If not Chrome, are you willing to switch to Chrome?