In RobotC there is definitely a non-sleep function, but you'd have to keep the program running the entire time (shouldn't be a problem). If you combined that with a rechargeable battery (see the lego website; I'm on my phone so it's hard to get links working, sorry), you could probably get it to work because the NXT can function while the battery is plugged in. The only problem would be that your robot wouldn't be mobile.
For activating it, you could use mattalen's suggestion, or you could get the Dexter Industries WiFi sensor and try to get it to work using that. Check Xander's website, botbench.com - he used that sensor to set up a robot where anyone could input a motor speed on a webpage, which would then be sent to the NXT, which would then move its motors accordingly (the users could also see that actually happening on a live stream video of his robot; it was really cool). You could set it up so it activates some function or changes a variable to serve as a trigger.
And one more thing: what you want to do is pretty difficult, and might be very hard using NXT-G (I'm not even sure if the WiFi sensor is supported in that, but don't quote me). So I'd advice using either NXC, RobotC or something else like it. I personally use RobotC and like it a lot, and Dexter Industries posted a bunch of tutorials on how to use their WiFi sensor in RobotC last week, but it's not free (not too expensive either, but still not free).
- Leon Overweel (aka. dimastero/ dimasterooo)
My blog: http://www.leonoverweel.com