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

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.
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.