Is there any way to draw power from a cellphone?
Charging USB jack on most phones is input-only.  However, some modern smartphones and tablets have a function known as USB OTG which is USB On The Go.  This allows the USB jack to supply 5Vdc at enough current to operate a USB thumb drive.

This mode is triggered when pin pin 4 of the micro-USB plug is jumpered to ground (pin 5).

There can also be other functions associated with that pin 4 on the connector - one of those functions allows the smartphone or tablet to be charged at the same time that the data lines are being used as the USB host.  These functions are triggered when a specific resistor value is used between pin 4 and ground.

There is much information available via your favourite search engine.  One graphic diagram is USB OTG connection diagram