This is a complicated question, mainly because the theoretical framework found in Aristotle's Poetics doesn't really address minor characters. Aristotle's understanding of tragedy was based on classical Greek plays which had a single protagonist (which would be Othello in this play), an antagonist (Iago), minor characters (everyone else), and a chorus. Aristotle's description of the action of a play (hamartia, reversal, recognition scene) all applied to events happening to the protagonist. Thus one can only apply Aristotle to the character Emilia by doing an analysis of what she contributes to the narrative of Othello, or by moving beyond Aristotle's Poetics and using his Rhetoric to analyze the persuasive nature of Emilia's speeches.
The Greek term "hamartia" is a noun deriving from the verb "hamartein" meaning to miss a mark, as in archery when an arrow misses. There is a sense of a mistake which is irrevocable, like an arrow released that cannot be put back in a bow. In the case of Othello, it is the belief that Desdemona is unfaithful and the decision to act upon that belief. What makes this a tragic character flaw is that (1) it happens to an otherwise great or admirable character and (2) it flies in the face of advice from other characters, and the protagonist has many opportunities to turn back from this course before the irrevocable action. Emilia is one of the characters who tries to persuade Othello to turn his back on his jealousy and trust his wife:
I durst, my lord, to wager she is honest,
Lay down my soul at stake: if you think other,
Remove your thought; it doth abuse your bosom. ...
Even though Othello trusts Emilia's honesty and moral character at this point, he believes that she is not very smart ("a simple bawd") and herself has been duped.
Emilia's final speech in Act IV shows her not, in fact, to be simple, but to be clear-sighted and intelligent, analyzing marriage from a cool logical perspective, using rhetorical devices mentioned in Aristotle's Rhetoric such as the topos of lesser and greater, inductive reasoning from examples, and reasoning by analogy.
Emilia's action in taking the handkerchief enables the castastrophe, the downfall of Othello. In the climax of the play, her main function is to prove Desdemona's innocence and Iago's guilt by revealing the true history of the handkerchief.
Rhetorically, what is most interesting about Emilia's lines in the final scene is that although she does evoke pity for Desdemona, she actually uses arguments from logos even more than pathetic arguments. Rather than being a "simple bawd" she shows herself an intelligent woman with a great deal of common sense, despite having been temporarily deceived by the cunning Iago.