Is there a way Apple support can do it on their end?
No, unfortunately Apple will not help regarding lost or stolen iPhones, just as Toyota would not help when a friend's SUV was stolen.
I was able to track it once the person who has it turned it on but they turned it back off.
You won't be able to "find" the phone if it is off, the battery has run down, or it is not currently connected to the Internet.
Your probably already have this information, but just in case........
If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support