You would not believe how many times I’ve seen as misused or used in a confusing/unclear way. Does it mean while here, or does it mean because? It looks like it means while, but who knows? (Work for a while with writers who are careless about words, and you’ll become paranoid about this sort of thing, too.)
At any rate, the sentence is compound and needs a comma after ended. Also, one to caused is wrong, obviously. Did the writer mean one to have caused? It’s correct but awkward, so I’m going with one who caused instead.
The world has ended, and while you’re trying to survive with your sister, you slowly start to suspect that she may have been the one who caused the apocalypse.
Perhaps it would work better to say you begin to suspect rather than you slowly start to suspect.