To ascertain whether John Updike's short story, "A & P," is a initiation story, a definition would be of help. Generally I refer to the story as a "rite of passage" story, but it is, technically, an initiation story. Mordecai Marcus, in his article "What is an Initiation Story?" provides us with the basic history of the initiation story:

The name and analytic concept of the initiation story derive basically from anthropology. The most important rites of most primitive cultures center around the passage from childhood to adolescence to maturity and full membership in adult society. Anthropologists call these rites initiation...

The author then goes on to describe how fiction can reflect the concept of such a ritual, in terms of literature. Alexandra Pulme cites Marcus' work in the following:

In his work about stories of initiation, Marcus offers a provisional working definition which contains the above mentioned aspects.

An initiation story may be said to show its young protagonist experiencing a significant change of knowledge about the world or himself, or a change of character, or of both, and this change must point or lead him towards an adult world. It may or may not contain some form of ritual, but it should give some evidence that the change is at least likely to have permanent effects.

In essence, then, the definition of an initiation story is one that shows a young character taking a step or steps that lead him (or her) "toward an adult world," with effects which will be permanent, life-changing.

In the story "A & P," Sammy is working in a grocery store. He is an adolescent who reacts as one would expect a young man to, when three adolescent girls enter the store dressed in bathing suits—his jaw drops and his eyes glaze over. One of the girls, who he refers to as "Queenie," has the beauty, poise and self-confidence (in his eyes) of a queen. However, when the manager criticizes the girls for their attire, the focus shifts from these secondary characters to the change that Sammy goes through within an instant.

Updike's young character takes exception to the manager's comments, though the manager is simply doing his job. Like a knight to the rescue of a beautiful princess in an Arthurian legend, Sammy defends the young girls, though they have already left the store. Sammy disagrees with his boss and then quits his job. His boss, knowing that Sammy is making a mistake to take exception to such a little thing, tries to reason with Sammy, but the boy has made up his mind. In the process, Sammy realizes that somehow something within him has changed: he is a different person than he was when he arrived at work that day. He knows the world has changed for him.

...my stomach kind of fell as I felt how hard the world was going to be to me hereafter.