Legality might be the last thing you'd think about. Laws vary from state to state.
She was told to stop and get a death certificate.
I find it 'very' hard to believe that any doctor is going to give a death certificate without removing/seeing the body from the RV to make sure there was no foul play and that he did indeed die of natural causes. I think there is way more to this story on 'how' it was done than this short clip of a story.
Transporting a family member AFTER they die in a hospital, a death certificate is issued after viewed by a doctor, from a funeral home with paperwork is done all the time and is legal.
Get stopped by the cops with a deceased person in your car without the paperwork for it and well I think you'd find yourself on the 10 o'clock news and in jail. Death in the family or not I certainly would not have done it.