Waverly tells her story from her point of view, making her the central narrator. That's pretty normal for stories like this; it helps us sympathize with her predicament and feel her pain. There's something subtle in the narrative tone, though, that makes the story a little more interesting.
See, as a character Waverly's a little girl, but she's not writing like a little girl. Let's face it: no nine-year-old girl uses phrases like "an impatient gathering of gurgling pigeons" (5) or says things like "One of the Chinese parishioners had donned a Santa Claus costume" (13). We can probably assume that Waverly is writing this as a grown-up remembering her childhood. This gives her wisdom that she couldn't have at the time the story takes place, and lets her see things—like the real identity of Santa Claus—that her kid self wouldn't.
This fits in with the tone of The Joy Luck Club itself, which uses lots of different narrators with lots of wildly differing points of view. Waverly's first-person voice is just one way of looking at the world, and the older Waverly's voice differs from this, bringing wisdom and perspective to the themes in the story.