Great questions. Registration would depend on the state requirements. Since your home of record is New York, here is a link to New York specific DMV recommendations. This will lay out your options to register the vehicle even when purchasing in another state.
Whether to wait to purchase your vehicle in Virginia is more of a personal preference. Is this for a PCS? Do you need it to drive some of your belongings? (Those are some decisions to make before purchasing- military moves don't typically include vehicle shipping- so you would need a way to get it there!)
Regarding the car buying service, it would depend on the dealership you are purchasing through. Once you apply for a loan on usaa.com, USAA can advise if you are approved and the terms of the loan. Then, you would take that information to the dealer and go from there.
I hope this helps answer some of your questions! Keep us posted on what you decide and good luck in Fort Bragg!