Powering an Arduino directly from a motor driver (DRV8825) or external power source?
why exactly is there only one power supply being shown here, is it because the Arduino can get its power from the driver?


No. It is assumed that you will provide the power to the Arduino yourself - either through the USB connection to the computer, or through the barrel jack.


  The thing I want to know, is whether this circuit would function without an external power supply for the Arduino or not?


No. The Arduino requires power to be provided.


  And if not, would connecting the Arduino to the power supply via the VIN pin work?


Yes, but it's not ideal.  At 12V the power wasted by the on-board linear voltage regulator will be proportionally high.  As long as all that is being powered is the Arduino itself that shouldn't be too bad, but if you try to draw any significant current either out of the 5V pin or through multiple IO pins (for example to drive lots of LEDs) then it will overheat and potentially shut down.

A more ideal arrangement would be to connect a 5V switching regulator to the 12V battery and use that to power the Arduino either through the 5V pin or the USB connector. Personally, I use cheap UBEC modules from RC model suppliers. They are small, low-cost, and very easy to wire up.