Have them review and format the content in filemaker before it is being sent out. That's the easiest way. FileMaker text fields support rich text formatting with the ability to get those contents as HTML.
Otherwise you would need something like the outlook manipulator plugin to do that, and even then, it's questionable at best if you'd actually be able to make the unsent email pop up from outlook. It may go to the drafts folder instead. Outlook has been getting more restrictive about external programs accessing it's functions.
1) Create an emails table in filemaker, with fields for subject, body, etc..
2) Create a layout that allows people to edit contents of email
3) Script creation of email record in filemaker and population of fields
4) Script button that takes contents of current email record and sends it.
Hmmmm, you see, we use TheBAT email client, so the Outlook Manipulator isn't really an option either way.
Maybe you're right... realistically, if ALL the manipulation, including from addresses/signatures/attachments, etc. can be done in Filemaker, then it really doesn't matter if the email client sees it or not... We're on IMAP, so the sent message will appear in the right box later anyway.
That said, even saving a new message into the drafts would be fine - the main use is for users to create emails for clients, which are then saved in the drafts for checking by a manager before being sent.