The track request never arrived at my child’s phone. What happened?

Of course you must have mobile data services enabled. Otherwise your phone cannot receive any requests when outside of Wi-Fi range. Please note that even within Wi-Fi range many devices switch Wi-Fi connectivity off a short time after the device display is switched off.
On phones using Android 2.3.x possible problem is that Background Synchronization is switched off on the phone. Check on Me relies on this feature to push data to the tracked and tracking phone. To enable Background Synchronization please open the phone settings app and check the settings under Accounts & Sync.