Of course there is a push! It just depends what you mean by pushing. The vaulter tries to "push" the pole towards vertical at takeoff. This energy is transferred through the hands. I'd like to say, that this energy is located in the lower chest, and the best way to maintain a proper pathway or linkage of the energy to the top hand, is to "push" as if you were in a handstand. Once you have swung to the cord of the pole, hang on.
Now, some might say, that the energy transferred from run-up and take-off through the top hand, and that this force is "pushing" the pole, but that's semantics.
I sometimes use the analogy of pushing a car. The most efficient way of moving the car, is to keep pushing. IOW, don't waste time going from pushing to pulling. You stay behind the car, and make sure the work you are doing, is keeping the car in motion.
Push, pull, row..... It all depends what you mean. I would argue, that Bubka did all of that at some point in a vault.