Looks like semantics is the problem here.
Your initial question was not specific enough to obtain a meaningful response. Yes, the vapor pressure does aid in pushing the liquid to the pump; however, this is usually nullified by the vapor pressure of the fluid. Remember this is a saturated fluid that will start to boil as soons as it encounters less pressure that its equilibrium storage pressure that you mentioned. This often happens as a fluid enters the suction side of a pump or compressor. So the only extra pressure that is available is the level of the fluid above the pump. This extra pressure is needed to ensure that fluid stays liquid inside the pump. The vapor pressure does push the fluid but it does not always push it in liquid form.