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Thanks for sharing with this issue.
As your description, the document list only have the“Read Only” permission. You can create a “Shared Mailbox” instead of document list if you want to synchronize emails that in the Outlook to MOP（Microsoft Online Portal）.
A shared mailbox is a mailbox that multiple users can open to read and send e-mail messages. Shared mailboxes allow a group of users to view and send e-mail from a common mailbox. Please refer the following article to set up Shared Mailbox in ECP.
Please feel free to let me know result and have a nice day!
Do you need any other help with this issue?
Thanks, I've create a shared mailbox on exchange, how can I access it on OWA?
Thanks for your reply.
You can find a method to open another mailbox from following URL.
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