Hi, Jill Taylor (JTaylor)
Good day!
Thank you for posting to Microsoft Community. We are happy to assist you.
According to your description, the issue of your concern that you would like to export the user's mailbox.
For disabling an account, please refer - Remove a former employee and secure data.
To backup the data as PST files, please give an attempt for the official instructions -Export or backup email, contacts, and calendar to an Outlook .pst file.
Also, could you please tell us whether you're using Exchange online?
If so, either manually exporting from Admin Center or using Power Shell can achieve it - Procedures for mailbox exports to .pst files in Exchange Server.
Please feel free to contact us if you need further assistance.
Sincerely,
Alex | Microsoft Community Moderator.