Can Exchange emails be downloaded in Apple Mail to then be uploaded to Gmail account? I use Apple Mail and I have a work account on Exchange/Office 365 that I use through Apple Mail. I do NOT have the Exchange password, I am not the administrator. Is there a way for me to archive or download all the emails (to a .OLM or .PST file?) and then import them to a Gmail account?

Bottom Line: I want my Exchange emails exported to a Gmail account.
Thanks for any suggestions!
If you don't own the account login details, you've picked the best solution, this would be to set up a rule forwarding all new emails to your Gmail email.

Any Mail Client, including Apple Mail, app needs log credentials to establish the connection. So to be able to use Gmail as client app for your Exchange account, you will either have 


to submit those manually  or 
to have the password in iCloud
keychain


For the older emails, you can export them in .pst and upload that to Gmail but that's rather desktop operation. I think this should help.