I've been searching around to no avail. I would really appreciate anyone who can point me to a place where I can download the background sound effect file for sigil stones. The one that sounds like stones being ground together. I don't have the game installed any more, and am kind of surprised I can't seem to find the file anywhere online. I couldn't even find a youtube video that was just that sound.

If you own the game, you have access to all the files. No need to download them from anywhere, because you already own them.
They will be in the BSA archives.

You can't find the files being shared online because that breaks the EULA.