Hi Robert,
This can be achieved by setting the 'forum_replytouser' setting. This is found in Site administration => Plugins => Activity modules => Forum.
This will mean that all forum mails come from the user themselves, and will include their e-mail address.
To then remove the e-mail address your users will need to change their preference for mail display to 'Hide my e-mail address from everyone'. You can change the default value of this setting for new users in Site administration => Users => User default preferences but this will not affect existing users.
To modify for existing users I believe you will need to make a one-off change of the data (e.g. in the database).
Hope this helps,
Andrew