I don't remember if this was a short story or just part of a novel. When I was young, the librarian at my elementary school would often read to us. He read a story about people, I think just one family, who were living in an underground facility after an alien attack. The attack had destroyed all carbon on the surface of the Earth. The story ended, or he stopped reading to us, right when the people living in the facility had made contact with other survivors using a shortwave radio.
This sounds like the first half of Jean E. Karl's book Strange Tomorrow: Janie, her younger brother and their father are in an underground bunker checking the supplies - it's a US government bunker for emergencies - when aliens destroy every carbon-based life form on Earth. Janie and her brother try to carry on while their father retreats into depression. They do theorize that there could be other survivors - in caves, underground, etc., and start listening on the shortwave for other survivors. It does end with their father and brother going to meet other survivors.