My children are dual US-Thai citizens with US passports, so that's no issue. It was just the wife who needed a visa.
For the emergency appointment application, we uploaded a photo of my dad's birth certificate and my birth certificate to prove relation, a letter from the funeral director with their contact information and the date of the funeral, and also a letter from my mother (this was not required) saying how much my wife has helped through this whole process and how she should be at the funeral.
Also, when we were dealing with the death and remains, the US embassy was extremely helpful. My wife was in contact with a Thai woman at the embassy through the whole process. She would not assist in any way with the visa process though.
EDIT: Also, I should add that we did not book our airline tickets yet and that wasn't an issue. We booked our tickets after she was approved.