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Dear Samantha,
Good day! Thank you for posting to Microsoft Community. We are happy to help you.
I went through your question, and it seems you have set up multi-factor authentication, but you can still access folders shared from External source, with out any attempt for approval!
Have you just set it up? some times such actions may take time to successfully apply. You may need to wait. Also, as mentioned by NoOneCan, please try to disable and wait for sometime to re-enable the multi-factor authentication.
If the issue persists after trying that, as it is urgent as mentioned, I kindly request you contact Microsoft Support directly so that they can give you a remote session, and see what could be going wrong on your side.
If you are not Admin, please contact your IT Admin to help you raise a ticket by following steps in this article: Get support - Microsoft 365 admin | Microsoft Docs. I am sure our Expert Engineers will effectively and accurately address your issue.
Thanks for your patience and kind understanding!
Sincerely,
Edwine | Microsoft Community Moderator