It's straight forward to add a web page as a web application. Find here a nice article.
The issue know is keeping a session open. Once switching to another app the session is closed and the user needs to re-enter his credentials.
I've found a similar entry without answer in stack-overflow.
Some hints ?
The trick is to do this:
I tested this to work on iOS 4.2.1, 5.1.1, 6.0 and 6.1. The session is even restored after turning off and restarting the device.
For a more elaborate discussion of this strategy you can take a look at my answer of this question: Maintain PHP Session in web app on iPhone