Is it possible for patella femoral syndrome to be caused by jumping on your feet without bending your knees?

Usually, PTFS is: A result of a biomechanical abnormality where there's a high angle between your leg and thigh (q angle) resulting in an increased amount of stress on the patella and patellar tendon.

Yes: The pressure of jumping without bending your knees will force the locking of the patella against the tibia bone. This will cause a shock effect to the tibia, patella and the femur leading to your pain.