Although quadcopters boast impressive traversal capabilities enabled by their omnidirectional maneuverability, the need for continuous pilot control in complex environments impedes their application in GNSS and telemetry-denied scenarios. To this end, we propose a novel sensorimotor policy that uses stereo-vision depth and visual-inertial odometry (VIO) to autonomously navigate through obstacles in an unknown environment to reach a goal point. The policy is comprised of a pre-trained autoencoder as the perception head followed by a planning and control LSTM network which outputs velocity commands that can be followed by an off-the-shelf commercial drone. We leverage reinforcement and privileged learning paradigms to train the policy in simulation through a two-stage process: 1) initial training with optimal trajectories generated by a global motion planner acting as a supervisory backbone, 2) further fine-tuning in a curriculum environment. To bridge the sim-to-real gap, we employ domain randomization and reward shaping to create a policy that is both robust to noise and domain shift. In outdoor experiments, our approach achieves successful zero-shot transfer to both obstacle environments and a drone platform that were never encountered during training.