Posttraumatic stress disorder due to natural disasters?

Can happen. Ptsd can result from witnessing or being victim of a natural disaster, especially if the emotional trauma is overwhelming and therefore hard to integrate into everyday experience. Ptsd doesn't always happen, though: two people can have the same traumatic experience and only one gets ptsd. The causes of ptsd are not completely understood, & are likely some mixture of genes and experience.
Definitely. A natural disaster where you were at risk of your life could very well cause PTSD. Disasters are by definition events that can cause us stress and harm. See a professional for assistance--but recognize that PTSD can start right away OR be delayed by months or years. Nightmares, avoidance or reminders, and a loss of sense of security are indicators you need help.
Repost your question. any horrifying or psychologically damaging event can lead to PTSD.