Which of the organelles in the animal cell is involved in exocytosis?

Golgi Body.

It's essentially the processing center of the cell. Products enter through the Cis end and exit through the Trans end in vesicles. The inside of the vesicles have the same glycoproteins, carbohydrates, etc as the outside of the cell, so when the vesicle reaches the membrane it can open up, release its product, and become part of the membrane.