Who is Emerson's intended audience in "Self-Reliance"?

Emerson's essay is aimed at young men--and we must emphasize men--graduating from college or in other ways about to embark on a life of opportunity. In other words, he writes to persuade privileged white American males like himself.
Emerson, like all of us, has blind spots because of the times in which he lived. While he was a strong abolitionist, for example, he did carry with him the racism of his time period, once likening blacks to animals ("elephants") and seeing them as having less reasoning capacity than whites. While he did embrace women intellectuals, such as Margaret Fuller, and would have wanted women to develop their God-given destinies, the use of pronouns and the assumption that his readers are deciding on a career (at a time when the vast majority of white women were homemakers) shows he was visualizing a male audience.
He assumes that his audience has the privilege of making choices about vocation and that their struggle is with the temptation of abandoning their destinies in order to conform to society and earn its rewards of wealth, honor, and power. Many people would not have this kind of choice to pick one path or another, but Emerson is not speaking to them.

In his day, Ralph Waldo Emerson was something like a rock star. Crowds came to listen to his lectures. Young people hearkened to his message that American society was on the verge of a new age; intellectuals responded to his philosophical ideas, and society in general responded to his optimism.
His essay "Self-Reliance" and its appeal to the young people was effected by his arguments that they were on the verge of a new age. In addition, many responded to his optimism and philosophical ideas, ideas about the relationships of humanity, nature, and God. Also popular was his belief that

[N]o government or church can explain a man’s heart to him, and so each individual must resist institutional authority.

Further, Emerson appealed to young people because he advocated acting from impulses, rather than rules, contending that the "self-reliant will triumph over the tyranny of time." This idea of trusting one's instincts and believing in one's unique talents and passions greatly appealed to many, especially the young, who felt that the nation—also young—was headed for greatness.