I have a 12V DC motor that I am using with an Uno and a TB6612FNG driver. As my 12V power supply, all I have is a 12V wall adapter that I can plug into the power input (not Vin). I was originally planning on powering the motor from the Vin pin when the adapter is plugged into the arduino. After some googling, it seems like this may not be a good idea because of the current that will be drawn from Vin, and I may fry the regulator.
Not the regulator. Look at the Uno schematic. There is a diode between the power jack and the input to the regulator which is also connected to Vin. It is a 1A diode. Many motors draw more than 1A when starting so could damage that diode.
Be careful when using the power plug or Vin. Powering through Vin or the power jack means that the Arduino and all peripherals that are on the 5V rail are powered by the onboard 5V regulator. The on board 5V regulator is not heat sinked so will supply limited current before it overheats and shuts down. The amount of current That the regulator can supply, safely, depends on the voltage input to Vin or the power jack. The higher the voltage the less current can by supplied. I would use a buck converter to drop the Higher voltage to 5V and connect that to the 5V on the Arduino, bypassing the, weak, 5V regulator. Then the rated current of the DC DC converter is available on the 5V line.