Do I need to store this in the fridge since there’s zucchini in it?
I don’t think these cupcakes are going to last long enough to be stored, and I would recommend serving them the same day they’re baked. If you do have to store them, I would recommend keeping them in the fridge in an air-tight container for no more than 5 days. Not only for the zucchini but for the cream cheese frosting too!