We present an alternative inference framework for the Gaussian process-based extended object tracking (GPEOT) models. The method provides an approximate solution to the Bayesian filtering problem in GPEOT by relying on a new measurement update, which we derive using variational Bayes techniques. The resulting algorithm effectively computes approximate posterior densities of the kinematic and the extent states. We conduct various experiments on simulated and real data and examine the performance compared with a reference method, which employs an extended Kalman filter for inference. The proposed algorithm significantly improves the accuracy of both the kinematic and the extent estimates and proves robust against model uncertainties.