While Emma performs the act of matchmaking (or tries to), the social convention of matchmaking seems to at the forefront of the text as well. After all, society dictates that certain pairings in the novel are appropriate while certain others are not. For example, social convention would deem Harriet's match with Mr. Martin appropriate, while it would probably frown on the one between Jane (an orphan and future governess) and Frank Churchill (which is why Frank keeps the match a secret). An interesting question to ask is, "Is society any better at matchmaking than Emma is, or should they both get out of the way and let people marry as they wish?"
Well, seeing how the novel's main focus is around Emma's matchmaking, you can really start anywhere:
Miss Taylor and Mr. Martin (The book opens just after their wedding, and Emma proclaims that she has all the happiness in the world because she made the match four years ago.)
Harriet and Mr. Elton (Emma tries to set them up at the beginning of the novel, things end up going very wrong when Mr. Elton professes his love for Emma.)
Harriet and Mr. Martin (It's not really "matchmaking" per se, but Mr. Knightley often encourages this match throughout the novel.)
Emma, herself and Mr. Churchill (Although this isn't a direct example either, Emma once says that if she were to marry, it would be to Frank Churchill, whom she'd never met, because he seemed, by circumstance, to "quite belong to her". Consequently, Emma often flirts with Frank, despite her obvious certainty not to marry.)
Harriet and Mr. Churchill (Once Emma thinks that Harriet is fond of Frank Churchill, she acts in ways that portray Harriet the better, and thinks to herself how awful Harriet were to feel when Harriet finds out about the secret engagement.)
Emma and Mr. Knightley (This one is a bit of a stretch to call "matchmaking", but after Emma realizes her love of Mr Knightley, she puts herself forward in a romantic manner.)
It's probably too late to post the response, but I hope someone finds use of this!