No, an object cannot have a moving center of mass unless an external force is acting on it. According to the laws of physics, specifically Newton's first law of motion, an object at rest or in motion will continue to stay at rest or in uniform motion in a straight line unless acted upon by an external force.
The center of mass of an object is the point that represents the average position of all the mass of the object. It is the point around which the object's mass is balanced. For a system of particles or a rigid body, the center of mass remains stationary unless an external force is applied.
If an object has a net external force acting on it, it will accelerate, and its center of mass will move. However, the internal forces within the object will redistribute the mass in such a way that the center of mass will always move in a consistent manner, obeying the laws of motion.
In summary, while an object's center of mass can move if subjected to external forces, it cannot have a moving center of mass without any external force acting upon it.