'Bereit' may be the key term. No one yet seems to have mentioned that 'to prep' is just a short form for 'to prepare,' as in prep schools (which prepare students for university, AE, or for public schools, BE), or test prep courses, which help students prepare to take standardized tests.
So in this case, a prepper seems to be someone intent on preparing for the worst -- in this case, stockpiling for the nuclear winter or whatever. However, it's evidently not just someone who likes to be prepared, like a Boy Scout, but rather, someone who takes preparation to an extreme.
It seems reasonable to me that German reports would just translate the word literally first (as 'Vorbereiter,' perhaps, or 'jemand, der vorbereitet') and then explain the context separately.
However, 'doomsday' adds an element that probably shouldn't be lost in translation. I like several of the suggestions, but Weltuntergang is probably my favorite. Could you say Weltuntergang(s)vorsorger?