This sounds like some sort of a ticketing system. kayako allows some soft of functionality like this but then again... i think customising it to what you need will be really painful.
Depending on your hosting platform, you can configure some kind of direct email pipe thing, i.e. once the email hit your mails server, it triggers a php script. That was how I got kayako working on my site... but then again.. it is highly dependant on your mail server.
From there.. u can slowly works towards the rest of the features.. which is basically just a php web application.