In Oklahoma there is an official court order that can give a party restricted guardianship over a child to seek medical treatment in the event of an emergency. It can be written to include anything both parties want. It is a court order and can be rescinded anytime either party wants.
It is relatively new and is official. If your child is hurt and needs medical care a paper will not make any difference what so ever if it is not an official decree.
The hospital will call a judge for permission to treat. If the judge gives permission they will treat your child, if the party with the child disagrees with the doc as to what is needed the doc will call the judge again and get another permission to treat for what they want to do, the person will have no say whatsoever.
It is just much safer and much more to your liking to have an official paper stating who has authority in your absence to make medical decisions for your child, it is safer to have that be an official court order than any letter or note from you, those could have been written in the car by a kidnapper as far as an ER doc knows. They will not recognize them in any way.
If you know where they are going to be staying, for instance if they were going to Chicago you could call a local doc and/or hospital and ask them what they needed for your child to be treated then they could fax you the paper work. They would need it notarized and then faxed back. It would be on file at a local hospital and if the child was hurt they would use that hospital and it would already be on file.
I worked as a nanny for a MS RN and she traveled frequently. She signed notes at the docs office listing us and at the local hospital in case the kids needed any kind of treatment. Then we carried copies of it all the time. That way if we borrowed the kids to take them to the drive in so mom and dad could have date night, if one of the kids got hurt we could take them to the ER and mom and dad could get there later but the kids would already be in the room and receiving treatment.
It is better to be safe than sorry.
I would make sure to have something official so my child would not suffer while the ER doc was waiting for the judge to get off the bench and take a call.