If you are powering the Arduino with less than 7V via the Japan Jack the internal voltage regulator won't work correctly or not at all. Thus your sketch won't run and can't control the motor shield.
You COULD feed an external power supply directly into the 5V pin of the Arduino, but only if you would power it with regulated 5V!
But: As a general rule this is not recommended as you will bypass all internal protection mechanism!
If possible, I would use 6 x AAA, if you can live with the limited battery life due to the capacity.
Have a look at this information about powering Arduinos;