I reviewed a manuscript which used my work. I recommended acceptance with minor revisions. Only after submitting my review, my wife discovered that my name is misspelled once in the manuscript. I'm a young scholar who would very much like to have one more citation. I sent an email to the editorial office about this but they never answered. Question: is it ok if I contact the author via email to tell him that I saw his manuscript -not confirming or denying that I was a reviewer- and ask him to please correct that mistake in proofs? In my experience people not always catch those by themselves, and because I recommended minor revision I don't think I'll see the manuscript again.
That is probably a bad idea.
But you can ask the editor to make the request instead. They won't want typos in the final result.
In general, communication with authors should be through the editor. Not the "editorial office" but the editor who assigned you the paper. An individual.
You are actually owed a response for this since you spent effort on behalf of the journal.
I agree with the other answers that an email, or a followup email to the handling editor is appropriate. Alternatively, if a preprint exists this would offer "plausible deniability" when contacting the author themselves (in case the editor remains incommunicado).
Be advised that a typo in a "citation" in the manuscript text along the lines of "user354948 et al. found that this discombobulation method did not reliably discombobulate" would not generally matter in terms of having your citation counted. Only the reference in the appendix is counted, e.g. by Google Scholar.
As an added point that has not yet been addressed: You mention your wife noticed the typo in the draft for peer review. Since you yourself state you are an early career scientist, I would just note for the future that peer review is generally supposed to happen strictly confidentially. Showing manuscripts you are peer reviewing to other people without the editor's consent is usually inappropriate.
If you saw their manuscript, you either are affiliated with the journal and in a better place to make the correction than they are, you're a reviewer, or a reviewer leaked the paper to you (and therefore they breached confidentiality).
Option one doesn't make much sense and doesn't apply. Option three is a pretty serious accusation that puts your colleagues under suspicion. The author may even feel the need to share this with the editor so they can identify the person responsible. Basically, either the authors should assume you're a reviewer or they should be ticked off.
Go through the journal instead. It seems you've already done this, and though it would be nice for them to have replied I'd note that they don't actually have to reply to you to make the correction. It's probably in their best interests to keep people happy who are willing to review for them. It's probably okay to send a follow-up email if some time has passed.
Note also that a misspelling likely doesn't remove attribution of credit to you; citation information is redundant by design, and your work should be findable with an incorrect or absent name.
I wouldn't do anything. It's just one citation. Remember, the number of citations looks impressive in your CV only when they are many. You shouldn't care much about one citation.
Besides, the typo may be found and corrected in the publication process. I remember receiving an article proof with a reference corrected by the publisher. I guess that publishers routinely check references for correctness and have automated systems for that.
Also, since the typo is only in your name, the reader will be able to find the referenced paper anyway by the bibliographical data provided in the reference (journal name, volume, page number, and publication year).
You say in a comment below, "I emailed the managing editor," so my understanding is that you are considering a further action. I don't think that a further action will really harm, but it may be seen as weird or embarrassing, especially if you contact the author.
My advice is: focus on doing great research. Think how you can advance science by doing something really important and useful. Think how you can write a paper that will be cited hundreds of times. Think big. You shouldn't waste your time and energy worrying about petty things like a single citation. Remember, your success will be determined by your mentality.
It looks like I have to clarify the point of my answer. If you want to become a great scientist, you have to learn one important skill - the skill to ignore unimportant things. Otherwise they will eat up your time and energy. Sure, you can now fix the issue with the citation, but don't become obsessed by fixing things like this.