If it was sent through the web interface - no.
The header information will only display the IP address of the server that sent it. There is nothing in the RFC that talks about interaction with web based mail clients. So, the system will just send it off to the SMTP server blindly and the SMTP server will say that the originating IP is coming from the IP of the host that the web mail is running on.
If the user used a client and SMTPed the email - yes.
When you use a client and send mail directly to SMTP, your IP will (should) be in the header as the originator.
As an example I sent an email to myself using outlook (I am using Google Apps):
Received: from ceres (xxx.net [99.144.xx.xx]) <-- This was MY IP. by mx.google.com with ESMTPS id 9sm1605084ywe.56.2009.10.31.17.27.00 (version=SSLv3 cipher=RC4-MD5); Sat, 31 Oct 2009 17:27:00 -0700 (PDT)