To answer the question, the best method to contact Delta Airlines is via phone. The phone agents are empowered to address emergency situations. Contact via phone can be accomplished via a number of methods from a foreign countries including Skype, USA phone with international calling, local phone with international calling or coordination with the hotel for options to call USA. Email is not a good option to address emergency situations.

Emergencies are best addressed via supporting documentation such as a death notice or letter from a doctor. Unknown if the staff at the airport are Delta Airlines staff or contracted staff so arrival at the airport without prior contact with Delta would be best addressed with the expectation of the purchase of a new ticket. Delta does have a direct flight to Atlanta but unclear if the destination is Atlanta or otherwise.

And please do coordinate with the inlaws on options for phone service such as a local phone or Skype. Quito is a city of 2 million people with excellent cell coverage and Internet.