Origami-inspired robots offer rapid, accessible design and manufacture with diverse functionalities. In particular, origami robots without conventional electronics have the unique advantage of functioning in extreme environments such as ones with high radiation or large magnetic fields. However, the absence of sophisticated control systems limits these robots to simple autonomous behaviors. In our previous studies, we developed a printable, electronics-free, and self-sustained oscillator that generates simple complementary square-wave signals. Our study presents a quadrature oscillation system capable of generating four square-wave signals a quarter-cycle out of phase, enabling four distinct states. Such control signals are important in various engineering and robotics applications, such as orchestrating limb movements in bio-inspired robots. We demonstrate the practicality and value of this oscillation system by designing and constructing an origami crawling robot that utilizes the quadrature oscillator to achieve coordinated locomotion. Together, the oscillator and robot illustrate the potential for more complex control and functions in origami robotics, paving the way for more electronics-free, rapid-design origami robots with advanced autonomous behaviors.