Where does James Joyce use the word epiphany in A Portrait of the Artist as a Young Man?
Oddly enough, I don't recall ever seeing the word "epiphany" in Joyce's A Portrait of the Artist as a Young Man. His work is famous for its epiphanies, as is Dubliners, but I don't believe he actually uses the word.
That being said, Joyce writes epiphanies into many of his stories and Portrait is no exception. If you look closely, you will find that Stephen has an epiphany as each chapter closes. The epiphany is subtle and difficult to verbalize and there are many interpretations of these epiphanies, but they are most certainly there.
For example, at the end of Chapter 4, Stephen runs into a young woman on the beach. He doesn't talk to her, but he watches her and realizes a lot about his own life. He mind cries out:
To live, to err, to fall, to triumph, to recreate life out of life! A wild angel had appeared to him, the angel of mortal youth and beauty, an envoy from the fair courts of life, to throw open before him in an instant of ecstasy the gates of all the ways of error and glory. On and on and on and on!
Here, Stephen is realizing that he wants to be an artist and that he doesn't want to dedicate his life falsely to the Catholic church. He doesn't actually say this, but that is the result. This is his epiphany at the end of Chapter 4. Look at the ends of the other chapters for more!