In "The Ones Who Walk Away from Omelas," how important are the minor characters to the story's primary conflict? How do they support or oppose the protagonist/major characters? And how do they interact with the protagonist/major characters, and what effects do these interactions create?

First, I think, we have to establish what kind of conflict the story portrays. It is, as the other educator wrote, an unconventional story, certainly, and I would argue that the reader becomes the protagonist, as the narrator speaks in the second person, directly to us. She asks,

How can I tell you about the people of Omelas? They were not naive and happy children—though their children were, in fact, happy. They were mature, intelligent, passionate adults whose lives were not wretched. O miracle! but I wish I could describe it better. I wish I could convince you.

The narrator makes the reader the main character, eliciting and attempting to describe our reactions and feelings rather than any other major characters' in the story. When she finally tells us about the one thing she thinks would make this community credible, the miserable and tortured child, she asks,

Now do you believe in them?

She explains that some people who learn of this child leave the community and never return, and the story invites us to consider what we would do. In this way, LeGuin creates a conflict that exists within her reader. What would you do? What would I do? Would I stay and enjoy perfect happiness, knowing about the child, or would I feel compelled to leave, too? By including the reader in the story, constantly referencing us as "you," the conflict LeGuin creates is that of character vs. self.
This means that everyone else in the story is a minor character. The existence of the child, and our learning of its horrible existence, is what incites the conflict within ourselves. The other characters in the story, mostly the ones who choose to leave, invite us to consider what we would do in this situation. They don't necessarily "support or oppose" us but, rather, encourage us to reflect on ourselves and come to some determination about what our own course of action would be.

"The Ones Who Walk Away from Omelas" by Ursula K. Le Guin is not structured as a conventional short story and does not really have a protagonist or antagonist. It is not so much a story about individuals as a moral parable.
The only major individual character in the story is the child who is locked away in misery and squalor. We know very little about the child, not even its name or gender. It is described as follows:

It could be boy or a girl. It looks about six, but actually is nearly ten. It is feeble-minded. 

As this child does not develop over the story and is not individuated except as an emblem of misfortune, it is not a normal protagonist; nor is it an antagonist, even though the happiness of the city depends on the suffering of the child.
The other characters in the story are also not individuated as characters but are part of a sort of social scenery, engaged in pursuits such as sports and music. What is most important, though, is that all inhabitants of the city at a certain age are apprised of the existence of the child. Some react to this by developing compassion, especially for other children. The ones who walk away decide that this grand bargain -- a utopian city enabled by the suffering of a single child -- is in some way unbearable and leave the city, but we are not told their individual stories. 
When the people visit the child, they occasionally kick it or sometimes just look at it but never offer it a kind word--even when it pleads with them.