I am currently in the process of arriving to my first duty station and currently my wife and kids are located in New York. The issue is, according to strict New York laws, my wife cannot move with her daughter without court approval. I would like to know if there's more information regarding this and being an active duty military member, what can I do to ensure that process doesn't take a long time and the move is smooth?
I'm going to assume that your wife has an order from a New York divorce court that specifically says that she is not allowed to remove the child from the state of New York with the permission of the court. That isn't so much a product of a "strict New York law", as you say, it's a pretty standard stipulation to many divorces in every state where there are children involved.
There's nothing at all that you can do. No concessions will be made because of your military status. You have zero standing in any of this unless you have formally adopted the child in question.
This isn't about you. This is about the child and the ability of that child to maintain reasonable contact with the other parent in a stable and safe environment.
You should understand that many don't think it's a great idea to take a child from their usual civilian environment into a military lifestyle. Many young people enter the military and raise families within the military lifestyle and that most often works well. However, when children who are older are transferred into the life of the military family, they may not do as well and the court will consider all that as the court decides any custody issues.
The mother of the child must file with the appropriate court and seek a modification of the existing stipulation that restricts travel. The father who is sharing custody, paying child support and who likely has visitation rights and court assigned schedules may or may not object to any actions to remove the child from the current situation. The father has rights, you don't. The court will decide what to do with a modification of custody and visitation schedules now that mom has remarried and wants to move.
You should be aware that your situation isn't all that unusual. I've seen this happen any number of times as has your new command. If your spouse doesn't take care of this formally and long before she joins you and in the appropriate court, and you and she decide to just move the child along with you to your new duty station, if and when this is brought to the attention of your new command, it won't be news well received.
No duty station is happy when the new soldier shows up with baggage. The less your new command hears of this the better.
A stipulation that a child can't leave the state without the permission of the court is a legit court order. Removing the child from New York without the written permission of the court places the parent in jeopardy of being in contempt of court and penalties can be harsh. Take care of this before making any other moves. This is a good time to cross t's and dot i's to ensure you've got it done right.