Would it be possible to run an LCD through the arduino using a 9v battery?

Post links to your hardware

why not; just a matter of how long the lcd will display something - very much depending, what type of LCD, backlight and Arduino you are using - take a look into the datasheets, see, how much mA every device consumes - have a look at your 9V battery capacity (mAh) and you can calculate how long you can power your project with the battery.

If it is too less, you have to consider: a) to reduce power consumption (LCD backlight on/off, Arduno type...) and/or b) use a battery with higher capacity