@DDFletch and @Chris_Hofer I am sure the button can be turned off directly in the settings area but I don't know enough JavaScript to do it in that area. I was able to figure out how to prevent it though on the screen after that. Essentially this script disables the form that is presented that allows the user to pick "course" or "quiz" and then export. I also added a line so you can put any information you would like to share.
{document.getElementById("exporter_form").style.display = "none";}
$('#export_files').after(
"<p><b>Please contact insert office for information on course exports</p>"
);
If you add this to your theme it will remove it. I will say it removes it for everyone including admins. I have never been able to get role based JavaScipt to work although I am sure other members of the community could share what to add so admins would still have this ability.
This hopefully will get you started.
-Nick