I would be sending the outgoing emails using a script containing the send mail script step. As long as all of the email preparation/editing is done in Filemaker rather than in the email client software, it would also contain a set field step to save a concatenation of the fields used in the email as a new record into a related table.
You can receive email directly into Filemaker using a plugin, but it may be easier to allow those emails to flow into your email software and then copy and paste them into a portal field for your individual client.
Filemaker's email capabilities are rudimentary at best.
There are commercial plugins that address this.