For some odd reason the staff on my website don't have the ability to change their profile pictures. I have it set so they can.
That sounds like a permissions issues - you can use the permissions analyzer to determine what is preventing them.
What happens when they visit this page? http://yoursite.com/account/avatar Assuming your site is in the root directory. Do they receive a permissions error and if not what happens when they click the upload button?
If the final value is Yes for uploading an avatar and they receive permissions error then an add-on could be causing it, disable the add-ons and have them try again.
