According to your description, it seems that you are using Exchange email account when sending emails. As I know, when exchange users accidentally delete an item from Outlook mailbox, you can often recover it via Recover Deleted Items From Server. So, please try to check your Recover Deleted Items From Server feature and see if you can find those disappeared items.
To further confirm your issue, it is suggested to log in to your web mail and see if you can find those disappeared emails.
User sends mass emails from the Third Party Program as invoices.
Does this issue only happen to emails sent from your Third Party Program?
To check if the issue is related to your Outlook client or your email account, it is suggested to exit other programs ,run Outlook alone and send test emails to see if this issue can be reproduced. You could also send emails from the web mail and see if there is any difference.
If this issue can also be reproduced when directly sending emails from your Outlook client, I'm afraid that the issue might be mostly related to your Outlook client. As I know, there could be some add-ins in your Outlook client that cause this issue, so it is suggested to start your Outlook in safe mode (Press Win + R, type “outlook /safe”, press Enter.) and see if the issue continues.
If Outlook can work fine without other programs, I'm afraid that the issue might be mostly related to your Third Party Program. Please try to troubleshot from the perspective of the Third Party Program and see if could helps.