The rated torque for the PG71 gearmotor is 1.5Nm. Manufacturers' rated figures are usually those that the motor can handle continuously, and are therefore much lower than the peak figures. They are what the motor could be run at all day long without overheating.
The 6Nm that you've calculated is needed to support this shooter is 4 times the rated torque, so the motor will definitely heat up
. The question is how long will it take the motor to get to an unacceptable temperature. A clue to the answer is the relatively short thermal time constant that this motor will have - it is a small motor. The brush and armature assembly will also be small, and these are the parts that will heat up and fry. Small means they will heat up fast.
As others have said above, the motor is stationary in this application with no circulating air cooling, the rated torque under these conditions will be even lower than the manufacturer's quoted figure of 1.5Nm.
The other thing to consider is the dynamic loading (as the robot bobs up and down and accelerates) that will be present and must be added to your calculated 6Nm figure. Plus there is friction in the gearbox and shooter pivot mechanism. Plus there is the reaction force from the ball being ejected from the shooter. All up, the motor could be seeing double the static torque.
My guess is that it will work, but only for a limited time, then it will fry and go open circuit.
Graham.