Having jumped through the family court hoops a few times, I will explain. There is physical custody, meaning the child lives with you, and there is legal custody, meaning you have the right to make, or participate in making, legal decisions (education, medical, etc) for the child. Having legal custody doesn't mean the child lives with you, or primarily with you.
This father should have, long ago, established legal custody of this child. If he had done this he could have prevented the mother from moving the child out of state without his permission. Strangely enough, since there is no custody agreement, the mother, essentially, also has no custodial rights.
At this point the father is screwed. There is no custody agreement so the mother is within her legal rights to move the child anywhere she pleases. There are probably no courts that will block the move, or force them to move back if they have already left. In these economic times very few judges are blocking moves that are based on job transfers. I know 2 fathers whose ex wives have moved their children, and both had legal visitation and custody agreements as defined in their divorce decrees.
The best thing for the father to do is to go to court, establish legal custody, and get a visitation schedule set. Since the father can prove that he has been involved, it is likely that his lawyer can, at the very least, get the mother to pay for travel expenses, since it was her move that caused the need to travel.