The solution was simple: use Microsoft's built-in Compatibility View. Direct your client to click the link to the right of the address bar that looks like a sheet of paper torn in half. After this, one of two things will happen: either the page will fix itself and show the client their messages, or you will get a "web page not found" error.
If you get the 404, direct your user to click the Back button and sign back in. They should then be able to access OWA without any further problems. Conversely, you can have them click "Basic" as opposed to "Premium" under the "Client" option on the log-in page. This should let them sign in using Compatibility View without the "web page not found" error.
