My ex lives in a different state. I have sole legal/physical custody of our children and there is no current visitation order. If a visitation schedule/order is sought out by my ex, is that handled in the state he resides in or in the state the children reside in?

The court’s jurisdiction is controlled by the location of the children. He would have to come to NC and file here. You should probably file and get an order for custody and child support in place.