If the net force on an object is zero, can the object be moving?
Acceleration is the change of velocity per unit time, so if there is no force, all we know is that the acceleration is zero. Therefore, the velocity is not changing. If the object was already moving, then it will just keep moving. So, yes, the object can be moving when there is no force applied to it.
Note: "force" in this discussion is to be interpreted as net force. Net force is the vector sum of all forces acting on the object.
Here, we have used Newton's 2nd law to show how it relates to his 1st law:
Newton's First Law of Motion: I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.