I know that the apostrophe can be used to denote the omission of letters in a word, so I'm wondering then if 'em can be used to denote the colloquial shorthand for him, or if it would be more proper to use 'im (and that 'em is really just shorthand for them).
I have only read or heard 'em used for them, though there is some disagreement in the comments (the discrepancy may be due to 'im and 'em sounding quite similar in speech).
You are correct that 'im can be used for him, and if you insist on butchering the language even further, you can use 'er for her.
However, keep in mind that these abbreviations typically aren't used in written language. The only case they should be used is when you want to emphasize the fact that an accent is being used. For instance, in Mark Twain's novels which take place in the South, those abbreviations are used when people are speaking, but are not used narratively.
In this example, the excerpt from the book The Adventures of Tom Sawyer is spoken, helping to emphasize the fact that Tom is speaking with the drawl and colloquialisms of Southern America:
“She! She never licks anybody – whacks ’em over the head with her thimble – and who cares for that, I’d like to know...”
While the narrative parts of the story do not use such colloquialisms:
The old lady pulled her spectacles down and looked over them about the room; then she put them up and looked out under them.
And for reference, a quote from the sequel, Adventures of Huckleberry Finn, abbreviating him:
“Yes, Mars Sid, A dog. Cur'us dog, too. Does you want to go en look at ’im?”
Thank you to bib for providing the quotes.