In this paper we consider the problem of recovering a signal x of size N from noisy and compressed measurements y = A x + w of size M, where the measurement matrix A is right-orthogonally invariant (ROI). Vector Approximate Message Passing (VAMP) demonstrates great reconstruction results for even highly ill-conditioned matrices A in relatively few iterations. However, performing each iteration is challenging due to either computational or memory point of view. On the other hand, a recently proposed Conjugate Gradient (CG) Expectation Propagation (CG-EP) framework is able to sacrifice some performance for efficiency, but requires access to exact singular spectrum of A. In this work we develop a CG-VAMP algorithm that does not require such information, is feasible to implement and converges to the neighborhood of the original VAMP.