My previous issue with IE 7 has been resolved.
All I had to do was add the site to the “Local Intranet” zone (which it should’ve been anyway), and all seemed to work as it did previously.
A bit more investigation revealed a security setting named “Include local directory when uploading files to a server”.
As for other browsers? Well, since this is an intranet app, it’s not important for the time being.
But if I were to implement something, it would probably be a custom dialog box where the end user could paste in the path to their document on mapped drives (or somethiing similar depending on how much time I had).