We proposed a switchable beam steering device with cycloidal diffractive waveplate (CDW) for eye tracking in a virtual reality (VR) or augmented reality (AR) display system. Such a CDW diffracts the incident circularly polarized light to the first order with over 95% efficiency. To convert the input linearly polarized light to right-handed or left-handed circular polarization, we developed a broadband polarization switch consisting of a twisted nematic liquid crystal cell and an achromatic quarter-wave retardation film. By cascading 2-3 CDWs together, multiple diffraction angles can be achieved. To suppress the color dispersion, we proposed two approaches to obtain the same diffraction angle for red, green, and blue LEDs-based full color displays. Our device exhibits several advantages, such as high diffraction efficiency, fast response time, low power consumption, and low cost. It holds promise for the emerging VR/AR displays.