This paper considers online optimal control with affine constraints on the states and actions under linear dynamics with bounded random disturbances. The system dynamics and constraints are assumed to be known and time invariant but the convex stage cost functions change adversarially. To solve this problem, we propose Online Gradient Descent with Buffer Zones (OGD-BZ). Theoretically, we show that OGD-BZ with proper parameters can guarantee the system to satisfy all the constraints despite any admissible disturbances. Further, we investigate the policy regret of OGD-BZ, which compares OGD-BZ's performance with the performance of the optimal linear policy in hindsight. We show that OGD-BZ can achieve a policy regret upper bound that is square root of the horizon length multiplied by some logarithmic terms of the horizon length under proper algorithm parameters.