Yes, if you're lucky four sensors in series have little resistance and can pass sufficient voltage-current to energize your relay coil. Although the load-carrying contacts may be 24V rated, the coil contacts can have a different voltage range of activation.
Already several questions pop up. For one, detail of the twist lock mechanism. Example... size, easiest method to distinguish an unlocked mechanism. The cost at the inductive sensor link appears high, yet the mechanism might not even be massive enough to trigger the sensor. The sensor itself appears large and suited to detect a door.