Why aren't my delivered products counting toward a quest?

This is because the quest is asking you to “Collect X by delivering X Style Products,” not “Display and Deliver X.” First you must deliver the product of the correct style and then you must also collect the item that randomly falls from the delivered products.

Collectible items (such as Shopping Bags, Peace Signs, and Paparazzi Cameras) fall from products of their style when you click deliver. You can collect them by clicking deliver and hovering the mouse over the item when it falls. You can also collect these collectibles from giving Glitter to clients or boosting your neighbors' displays.

The longer your display duration, the more likely it is that you'll see a collectible fall. Good luck!