No. The iPhone cannot currently do anything like that. There was an early preview talk about a background push framework at Apple's WWDC in June 2008, but still hasn't been made real yet.

I appreciate the app author's desire to create something that can enhance the AMBER alert system, I think that this app will be useless until the iPhone supports background processing.

Yeah, and more importantly, can't you just sign up your phone (you know, with a phone number) for TXT or SMS alerts from the AMBER ALERT folks? I'm certain some of my family members have gotten AMBER ALERT TXTs before.

And then, you know, since your phone has internet access you could just pull up their webpage or local news through safari.

No. A notification API/service was supposed to be available to work around this. But no further (public) news about it has been released.

<incredulous>You can't even make an alarm clock app for the iPhone??</incredulous>

You can. But the user must keep the app running, i.e. cannot switch to another app or go to the homescreen. Switching off the screen (the power button), or answer (not initiating) phone calls is OK. Which is kind of a big limitation. But really has been OK for millions of users so far...