Model-based control faces fundamental challenges in partially-observable environments due to unmodeled obstacles. We propose an online learning and optimization method to identify and avoid unobserved obstacles online. Our method, Constraint Obeying Gaussian Implicit Surfaces (COGIS), infers contact data using a combination of visual input and state tracking, informed by predictions from a nominal dynamics model. We then fit a Gaussian process implicit surface (GPIS) to these data and refine the dataset through a novel method of enforcing constraints on the estimated surface. This allows us to design a Model Predictive Control (MPC) method that leverages the obstacle estimate to complete multiple manipulation tasks. By modeling the environment instead of attempting to directly adapt the dynamics, our method succeeds at both low-dimensional peg-in-hole tasks and high-dimensional deformable object manipulation tasks. Our method succeeds in 10/10 trials vs 1/10 for a baseline on a real-world cable manipulation task under partial observability of the environment.