Android - Using an OTG cable while charging
Yes, it is possible, at least technically. What you need is a cable called a OTG host cable with power, which is a double-female cable with a male micro, a female micro (which only has the power and ground connected, no data), and a female A, like this one.
Additionally, you also need software support. I know an appropriate kernel patch is available for CM10 on the Nexus 7 (here), but I don't know about any other devices.
It's possible, some phones support Battery Charging Specification 1.2 and can utilize USB ACA (Accessory charging adaptor) which support OTG + Charging at the same time. (the Acasis H027 is one such adaptor)
My phone (Motorola Moto G XT1032) is not able to auto-detect such an adaptor out of the box, but after modifying the kernel, I was able to charge it and use USB Flash Drive + USB Ethernet at the same time.