How do I ask a professor to contact another professor he knows to accept me as a Masters student *via email*? I did research with a professor during my senior year (2017/2018) and we published a paper together. He wrote me a letter of reference for my grad school applications. 

One of the professors I'm interested in working with at another school is his sister-in-law. I've tried emailing her to no avail. I had a friend suggest I have the professor I know email her instead, since even the fact that they're related aside, a professor is more likely to respond to an email by another prof.

This is identical to this question: How do I politely ask a professor to contact another professor he knows to accept me as PhD student?

However, the answer to the previous question suggests subtlely bringing it up during a conversation. I'm not in school anymore (just graduated) and don't meet this professor in person anymore so I need to do this via email.

I was hoping someone could give some advice about how to compose an email politely requesting him to contact his sister-in-law on my behalf? I feel fairly embarrassed having to do this since I am asking for quite a bit of him, but I figure it's worth a shot. Worst case he says no.
Since you aren't now his student and can't visit in person, I'd send a rather longish mail first requesting that he do you this favor, sending a letter to "Prof X", then giving the background that you've been unsuccessful and finally reminding the prof of the old work you did and any relevant updates. 

I think that making the request first will permit him to quit reading at any point so that the length of the mail is less important as normally a short mail is appreciated. 

"Longish" implies about a page with about 3 paragraphs, not a monograph. 

Somewhere along the way note that you really want to work with his relative and that you know of the relationship. But don't make that the main focus. Letter from Prof A to Prof X. 

But I don't think you are asking a lot. Part of his job is to send out such letters. It may be that sending it to a relative will cause him to send a different sort of letter, but perhaps not. The other prof may take a recommendation from a relative as having more weight (honesty) or not. But that is up to them.