I'd like to create an app that the user downloads for the purpose of having it
always run to monitor a specific activity.

I've seen threads with
comments like:

"That's a bad idea. If there was a way, then every app
would want to use it", but if a user knows an app is started when they turn
their iPad on, and they want it that way, why would that be a good thing?

I'm afraid this would not be possible. Apps on the iPad aren't allowed to interact with each other - they are isolated (Apple calls this 'sandboxed') so that a virus infection cannot affect the iPad. Your suggestion would require one app to be able to gain access to another apps activity - something that iOS, the iPad's operating system, doesn't allow.

Tim