"I live in New York. If I become a medical marijuana patient in New York, can I get my medicine if I’m out of the state?"
No. You or your designated caregiver must pick up the medical cannabis in person with proper documentation. According to § 1004.13 Security requirements for manufacturing and dispensing facilities (2)(o) "A registered organization shall only transport approved medical marijuana products from a manufacturing facility to dispensing facilities."
No. Cannabis is currently a Schedule I controlled substance. Transporting cannabis across state lines is illegal. It is also illegal to travel in to or out of the United States with cannabis in your possession.