The main objective of this presentation is to explore mathematical programs that incorporate data uncertainty in the vanishing constraints (UMPVC) and to solve them by using a robust optimization framework to deal with the worst-case scenario. To begin with, we derive robust Fritz-John conditions for the UMPVCs and introduce extended no nonzero abnormal multiplier constraint qualification to obtain robust Karush-Kuhn-Tucker conditions. We also identify the robust strong stationary points of the UMPVC and attain sufficient optimality conditions under generalized convexity assumptions. We also identify robust weak stationary points of the UMPVC using a tightened nonlinear programming approach to seek necessary and sufficient robust optimality conditions. The robust version of several constraint qualifications (CQ), like Abadie CQ, Mangasarian-Fromovitz CQ, and linearly independent CQ, are introduced to handle the uncertainties associated with the special structure of the vanishing constraints. Several algorithms are given to apply the results and various examples are presented to illustrate the algorithms.