[blind-eddie]

I to wanted this for a very long time & found many scripts that would do just what you want, but your members will still have to use their email user name & password to login.
The script I use offers a free lite version that will allow each person only one email address & a paid version that allows people to have many email accounts, just like outlook.

Both will be good for you, as I am sure you have a few @yoursite.com email address as I do.
You are allowed to use the paid version for free if only 5 users use it, (Unlimited email addresses per user) if more, then it gets expensive.
The lite version is just what you need. Adding a link to it via profile is easy enough, but the navbar would be a better place.
I have it set up on all my vbulletin sites for members to be able to check their myname@mysite.com emails.
I host a few sites & set it so anyone could use it within my domain space.
I have a stand alone version & the very same one added to the forum on its own page.

The links below are to the stand alone version & to the vbulletin page I added it to on my 3.8.4 play site. (I have others)

Stand Alone

vbulletin page

Look in the footer for link back to script author. Can't advertise paid products here.
If you use it & need help, feel free to contact me.