One possible reason for differences between messages viewed in Firefox would be the format the sender used. For example, HTML and plain text messages may behave differently. In HTML, the link actually is in the message sent by the sender. In plain text, there are no links, but the desktop version of Outlook finds them and makes them clickable. Perhaps the same is true of the "Premium" version of OWA served to IE. Other browsers have long been second class citizens...

Maybe there is an add-on that fills in the missing functionality?

There is usually a toolbar button in the editor to turn a text link into a clickable hyperlink (look for a chain like button).
You can select the text and click that button.
If you can't find the button then hover them all to check the tooltip of each.