The government will supposedly reject the application on the basis that the person is an illegal alien. (Although an Obama appointee recently allowed an illegal alien to buyi a gun, so go figure.)
So, theoretically, that reports to the Feds that we have an illegal alien buying a gun. Not sure if the ATF ever reports that to ICE, though. Probably depend on who is president.
Not sure about more specific reporting requirements by states. Seems like a good idea though, one that even Q could support. Gun shops usually try to keep good relations with local cops & sheriff's, so it might depend on the local authorities whether any additional reporting occurs.