Merriam-Webster defines an epiphany as:

a (1) : a usually sudden manifestation or perception of the essential nature or meaning of something (2) : an intuitive grasp of reality through something (as an event) usually simple and striking (3) : an illuminating discovery, realization, or disclosure 

b : a revealing scene or moment

There certainly is an epiphany in this remarkably poignant story in terms of all three of the above definitions. The epiphany occurs in the closing paragraph:

I took my grandmother’s regalia and walked outside. I knew that solitary yellow bead was part of me. I knew I was that yellow bead in part. Outside, I wrapped myself in my grandmother’s regalia and breathed her in. I stepped off the sidewalk and into the intersection. Pedestrians stopped. Cars stopped. The city stopped. They all watched me dance with my grandmother. I was my grandmother, dancing.

Throughout the story, the narrator describes his mundane, poverty-stricken life and tells us about the friends he has had and lost. He and many other Indians have become indigents. Their lives are informed by shallow associations with other Indians from different tribes who are, in a way, connected to one another through their dependence on alcohol, which provides them some form of escape from the misery that they face every day.

Throughout his encounters, the narrator is generally sardonic about Indians. Their circumstances, it seems, have unfortunately led others to form a biased assessment of them.

We’re common and boring, and you walk right on by us, with maybe a look of anger or disgust or even sadness at the terrible fate of the noble savage.

It is apparent, however, that the narrator does not really care much about others' opinions and only states, "But we have dreams and families." In the urban environment, it appears as if they have lost their identity, separately and communally. They are all deemed the same, irrespective of which tribe they belong to. The narrator, though, is aware of the differences and mentions them specifically. Ironically, however, it seems as if he has lost his identity, as he states:

“Piece by piece, I disappeared. I’ve been disappearing ever since.”

He is obviously also disillusioned with what he has done to himself after his grandmother's demise, as he informs officer Williams:

“I’ve been killing myself ever since she died.”

Others, though, recognize that the narrator is somehow different. They see that he is intelligent, for example, as seen in Officer Williams' and the waitress' observations: 

“Listen to you, Jackson. You’re so smart. Why the hell are you on the street?”

“O.K., Professor,” she said. “I’ll only ask you real questions from now on.”

In his quest to win back his grandmother's regalia, the narrator also makes it clear that he has been careless, but that he has now found something that he can really care about—something that he can truly identify with.

“That may be. But I care about it. It’s been a long time since I really cared about something.”

When the pawnbroker gives him his grandmother's regalia at the end of the story, the narrator's moment of illumination makes him realize a number of things:

Firstly, just as the bead in the regalia is used to indicate an imperfection, he does not need to be perfect. Secondly, the bead is unique, just as he is. He is not the same as anyone else, within or outside of his culture—he has an identity. Thirdly, although separated from his culture, he is still part of it, and it is within him—irrevocably and inalienably. Fourthly, he is connected to his history. He and his grandmother are cut from the same cloth. The good that he identified within her is, in the same way as his culture, a part of him. Her qualities, he realizes, have been passed onto him and, as such, he is in essence also her.

Though we do not learn what the long-term effect of this newfound perception could have been on the narrator, one can only assume that he would have been inspired so much that he would have gone forward and done better for himself.

I would think that the ending is a type of epiphany.  The lack of emotional connection that had defined Jackson for so long in the narrative is interrupted by the end.  At the moment he is dancing with his grandmother's regalia, an epiphany is reached:

Pedestrians stopped. Cars stopped. The city stopped. They all watched me dance with my grandmother. I was my grandmother, dancing.

This is an epiphany for a couple of reasons.  For one of the first times, Jackson is emotionally and spiritually connected with something in a positive way.  Jackson had been defined as one who had been unable to forge anything in way of emotional connection and understanding.  With his dancing, one sees that Jackson has understood the power of spiritual connection with something larger than himself.  This moment is an epiphany because of its nature of awakening.  At the same time, Jackson has understood the interlinked nature of being in the world, one that represents how interlinked individuals are.  The path towards getting to the point where Jackson is dancing and unified with his grandmother is one in which there were many individuals responsible for many different elements.  In this, Jackson once again understands how he is not alone and isolated from others, but rather recognizes his state of being as one in which individuals are indelibly linked to one another.