Optical coherence tomography (OCT) has found great success in ophthalmology where it plays a key role in screening and diagnostics. Clinical ophthalmic OCT systems are typically deployed as tabletop instruments that require chinrest stabilization and trained ophthalmic photographers to operate. These requirements preclude OCT diagnostics in bedbound or unconscious patients who cannot use a chinrest, and restrict OCT screening to ophthalmology offices. We present a robotically-aligned OCT scanner capable of automatic eye imaging without chinrests. The scanner features eye tracking from fixed-base RGB-D cameras for coarse and stereo pupil cameras for fine alignment, as well as galvanometer aiming for fast lateral tracking, reference arm adjustment for fast axial tracking, and a commercial robot arm for slow lateral and axial tracking. We demonstrate the system's performance autonomously aligning with stationary eyes, pursuing moving eyes, and tracking eyes undergoing physiologic motion. The system demonstrates sub-millimeter eye tracking accuracy, 12 μm lateral pupil tracking accuracy, 83.2 ms stabilization time following step disturbance, and 9.7 Hz tracking bandwidth.