It is hard to say how you have the addresses stored from the screen shot so I am making some assumptions but the concepts will stand. You have several check boxes that I assume are used for the email addresses. In the after update of those checks, do something like
(you will have to add some logic in to determine if the , should be there or not. Maybe a boolean firstEmail and if it is true don't use a comma otherwise put a comma before the address, just make sure to set it to false after the first time)
You would have to have this for every check box.
Somewhere you have all the email addresses, you could come up with a cleaner way of getting the correct address based on the check box chosen.
I would suggest that you store them in a table, with a numeric ID as a key, then create the check boxes as a group and set the values to match the appropriate key from the table. You could then do a query on the table where ID = emailToGroup.value whenever the group is updated. You would then have the email address and you could update your text field with it using the same basic idea as above. This way you do not duplicate any code.