Hi,
There is no included migration tool for migration from one Office 365 to Another.
My advice is that you do a PST-export in Outlook for the entire mailbox in your clients account. Then configure Outlook to your personal Office 365 account and then import the entire content of the PST-file to that mailbox - migration done.
You can use this article as reference: support.microsoft.com/.../287070
Hope this helps.