I would verify that their Exchange account is the only account within outlook. From a few different places it looks like it could be something stuck in Drafts too (although you said you checked there). Possibly removing the account and re-adding it in Outlook could be the quick fix to it (although they would lose their saved autocompletion which a lot of people complain when they lose it).
I am using Outlook 365 for Mac Version 16.42. Yesterday email attachments suddenly were not showing up when I opened email in Outlook. The paperclip icon shows on the list of emails. I am able to view attachments on my phone, with Apple mail on the Mac and using the web interface for my email. It is only Outlook that is causing a problem. When you install Outlook, an Inbox repair tool (scanpst.exe) is also installed on your PC. The Inbox repair tool can resolve problems by scanning your Outlook data files, and repairing errors. To use the Inbox repair tool, exit Outlook, and then follow the steps in Repair Outlook Data Files (.pst and.ost). 12 hours ago I am using the Outlook program in Office for Mac v16.49 (21050901) on a 27' 2017 iMac running Mac OS Big Sur 11.3.1 Recently (for the last couple of weeks or so) I've been unable to send emails that contain photo attachments. The files are JPEG's about 200 kb in size. When clicking the Send button, a warning dialog box will pop out to say that you may have forgotten to attach a file. See the following screen shot: Auto reminder when forgetting attachments in Outlook 2007 and 2010 Microsoft Outlook 2007 and 2010 do not support automatically warning if you may forget attachments. Attachments exist in GMAIL web interface but not in recent emails on Outlook for Mac. Source As apparent from the above comments, users facing this issue are unable to view their attachments at all via the Outlook app for Mac.
Outlook Opening Mail Attachment Warning
Im running 2016 on probably 25 Mac's and haven't seen this issue yet, but I have had weird issues with Outlook in general that aren't worth the time messing with and just adding the account back again will fix saving all of our times.