The book says that Daphne was on board a ship that was heading to Port Mercia to retrieve the crown. That was the cause of Daphne being on Nation throughout the book, but was it possible that this all could have happened without the plague?
The plague sets the scene for Daphne's princesshood.
Could the story have happened in the same way without the plague? Mostly, yes. The subplot with the Gentlemen of Last Resort would have been removed from the story, and Daphne would have been just a girl from the English gentry rather than an heir to the throne. The events on the island would have played out in pretty much the same way until the very end when the Gentlemen of Last Resort arrive.
So, what does this add to the story? Why was it written specifically to make her an unknowing princess?
It adds to the irony and contrast between her life in the Nation and the life she'd have in England. Much of the story is about this contrast: the cultural shock of putting a girl from an English noble family into a very different community, forcing her to learn their ways, learn to live among them, and to feel more at home there than she had in her old life. Ironically exacerbating this is the fact that her own society's expectations of her are even more different than she'd realised from what she's doing in the Nation. From their point of view - from her grandmother's point of view, for example - it's bad enough for an English noble girl to be spending time with Mau and his people, but even worse for a princess. We the readers are aware of this, but she isn't, and thus there's a heightened sense of irony and also of tension - what will happen when her family finds her, will they take her away, what will they think of her?
It makes the story's bittersweet ending more inevitable. Leading on from the last point in my previous paragraph, the consequences when her family arrives are indeed worse than she'd expected, from the point of view of her life with Mau in the Nation. If she'd had the same status as before in English society, she might have been able to continue her life there, her grandmother notwithstanding. But as the daughter of the new king, she was forced to go all the way to the other side of the world and live with her father in England. There was no choice, and that spelled the tragic end of her relationship with Mau.
One correction: you said that
The book says that Daphne was on board a ship that was heading to Port Mercia to retrieve the crown
But here you're getting confused between two ships. It was the Gentlemen of Last Resort who were going to Port Mercia with the crown to find the new king. Daphne was on the Sweet Judy, and knew nothing about the plague or the issues of royal descent. So the whole plague/royalty issue really is irrelevant to the actual events of the story (although affecting our perceptions of them) until the final chapters.