@tteh00 have you tried commanding the gripper without the "and_wait" option? If you removed that you can call open and close and have the robot moving at the same time.
You can either do it with the URCaps checkbox and_wait or with the script function rq_close (instead of rq_close_and_wait)
While the UR robot is moving from one position to another position, is there any way to trigger the Robotiq gripper to open/close ?
The usual way is the robot has to move to a point, then execute gripper to open/close, then move to another point.