One approach to improve the accuracy and robustness of vision-aided inertial navigation systems (VINS) that employ low-cost inertial sensors, is to obtain scale information from stereoscopic vision. Processing images from two cameras, however, is computationally expensive and increases latency. To address this limitation, in this work, a novel two-camera alternating-stereo VINS is presented. Specifically, the proposed system triggers the left-right cameras in an alternating fashion, estimates the poses corresponding to the left camera only, and introduces a linear interpolation model for processing the alternating right camera measurements. Although not a regular stereo system, the alternating visual observations when employing the proposed interpolation scheme, still provide scale information, as shown by analyzing the observability properties of the vision-only corresponding system. Finally, the performance gain, of the proposed algorithm over its monocular and stereo counterparts is assessed using various datasets.