Our site (http://www.rcu.msstate.edu) does not work if the "www" is omitted from the URL. I would like to set up some kind of redirect to make it so it will work. I followed these instructions (https://www.youtube.com/watch?v=XFQ_Gj9sBag) on changing the Site Alias settings from canonical to redirect, but it didn't help. Any ideas?

Oops, my link is bad due to the closing paren. Try this:
http://www.rcu.msstate.edu

Make sure that:

-- non-www is set as the primary alias
-- both non-www and www are bound to the site in IIS

Oh ... it looks like there is no DNS entry for rcu.msstate.edu. That is sure to cause a problem!

I don't know my way around IIS, but I'm told these are configurations are set, but that an old IT person had said for some reason, we couldn't do it.
Screenshot of site bindings: http://grab.by/G10o
Is this something that would need to be configured outside the server?

If you try to ping the rcu.msstate.edu, you will find that it does not have a DNS entry. You should fix that first. If IIS is configured correctly, and your website is bound to both URLs, then all will be well.