Simple.
1-create an iCloud email address and see that it registers in Apple Mail with its own inbox.
2-create a sub-mailbox for the iCloud mailbox by right-clicking the iCloud mailbox and select "new mailbox".
The new mailbox will appear at the extreme bottom left side of the window. Drag the box to the top of the window, above the other mailboxes and rename it whatever you like.
3- For each non-iCloud mailbox, select (highlight) any email in that mailbox, go to mail preferences, and add a
"new rule" (on the right side of the preference window). The new rule will be as follows:
-Use the drop-down menu to select "if any of the following conditions are met"
-Then select "to" "contains" "the email address (automatically filled in)"
-Then "Perform the following actions" "move message" to "the iCloud sub-mailbox you created"
So any non-iCloud email received by Apple Mail with be forwarded to the iCloud mailbox and can be seen as saved in the iCloud website.
I called Apple Support first- they said there was no way to store any emails other than "ICloud email addresses".
That is not true. I did this with 2 gmail and 1 ATT email addresses. It works perfectly.