In this paper, the Uncertain Lambert Problem is solved using higher order polynomial series instead of the conventional first order one used in linear analysis. Coefficients of the polynomial series are computed in a Jacobian free manner, providing a computationally tractable approach. This polynomial series is exploited to compute the density function for the Lambert solution given the probability density function for initial and final position vector. Non product quadrature method known as the Conjugate Unscented Transformation (CUT) approach is used to construct coefficients of polynomial series by solving a minimal number of Lambert problems through the intelligent sampling of the uncertain initial and final position vector space. Numerical simulation results are presented to validate the proposed approach.