Arduino is micro-controller and not a raw power source. It should NEVER be used to supply power to any peripherals or sensors because even though the voltage at the GPIO pins is 5V, the current is only limited to 50 mA per pin and total of 200 mA. Thus, having 12 SG90 servos connected to the Arduino is not possible.
You can use the 12 V battery and plug it into the 12V DC jack of the Arduino board to run the board. But, to power up the 12 servos, use the same 12V battery (assuming that it has a current capacity of around 3.0 - 4.0 A or more with a total of 5000 mAH capacity for enough duration) and connect it to a voltage regulator like LM 7805 to obtain 5V output for the servos. However, an LM 7805 IC has a current limitation of 1A and I found this FEETECH (Fitec) FS90 9G Mini Servo with Accessories with specifications for the motor mentioning that the servo could be pretty current greedy at times. My best call would be to use one voltage regulator for 2 servos each and you should be good to go.
PS - Do not forget to make Arduino's GROUND and the LM 7805 output GROUND common (not needed if the same battery source is used) otherwise the PWN signals sent from the Arduino to servos won't work.