He needs to file where Mom and the kids live.

If he and Mom can come to an agreement, they can file a stipulation with the court in Arizona; they'd likely not have to actually appear in front of the judge unless there was something irregular with the parenting plan.

Otherwise he'll either have to travel to Arizona or hire an attorney in AZ to appear on his behalf. He should start researching long-distance parenting plans, and think about including Skype/webcam visitation too as well as regular phone calls with the twins. He should be specific - such as "Mom will make the children available for phone contact on X and Y evenings between the hours of 6-7pm" (that's just an example).

He needs to do his best to iron out who is responsible for pick up and drop off, and determine when exactly vacation periods start and finish. When does he have to give notice for summer vacation? April 1 is a popular deadline. Things like this.

He can also request that Mom is responsible for half of the cost of transportation.

Unfortunately Mom is correct, at the moment - without a parenting plan in place she doesn't have to allow any visitation at all. So it's absolutely in Dad's best interest (and the interest of everyone, actually) to get a court order.

Court orders are our friends. They protect everyone involved