Two 300RPM 12V DC motors connected to arduino motor shield r3 power supply
The external power of the shield is connected to the VIN pin, so it will power the Arduino via that. (vin goes to the voltage regulator that will provide the 5v the arduino needs).

You can connect two batteries in parallel to provide mode current.

But since they are 12v motors, they would need 12v to run at maximum power. 9v is a lot less than that.