You should be anle to move. Check with your NJ probation officer/ office and/or probation office in NY. I can't speak to the automated reporting aspect. NY is the state in control of your probation. NJ to this point and PA if you move are only servicing NY's probation. Should be no problem but you need to clear it in advance.
You need to check with your probation officer first and get their approval to relocate. These requests are routinely granted so long as you otherwise have been a good probationer. Your officer will decide if the monthly reporting will be in person or by phone. Get approval before you act.
I am trying to give you a general answer to your question. We do not have an attorney-client relationship by this response on the avvo website. I have not been retained to represent you. I am licensed to practice law in Kentucky and in federal court in this state and the Southern District of Indiana. You need to seek legal advice from an attorney licensed to practice in your area..