[El_Muerte]

A webmail client is usually a very complex piece of software.
But there are a few approaches, but most importantly...

You must have rfc2822 clean usernames, a lot of ascii characters are not allowed in the name part.

Then there are these approaches:

1. redirect the received emails
This would redirect the email to the email address the member has registered in the forum.
This can easily be done with a procmail script and a shell script to retrieve the destination email address. And of course you need a catch all email address.

2. redirect emails to the PM inbox
An email to a given address would end up as a private message. Just like with the previous approach this can be done with a catch all email address and a procmail script and shell script. Of course this adds the limitation of not being able to send emails from the inbox. But that could be "fixed".
However, this approach does require various security checks and content conversion of the emails.

3. use vBulletin as authentication service
With this you use vBulletin as the source for authentication, an existing webmail client (for example IMP) needs to be extended to look up the user credentials.
However, this will require a tricky email server setup, which is probably not possible unless you have root access.