For instance, what is the equivalent of New Yorker when using the acronym (NY or N.Y. instead of New York)?
I doubt it's formalized, but this is akin to how one forms the plural of acronyms. For example, compact discs: CDs or CD's?
Since the apostrophe denotes possession, it's more often correct to omit it when merely pluralizing, though there are some corner cases:
The Chicago Manual of Style has an interesting way to address this: They omit the apostrophe, unless there are periods in the abbreviation. So this would give you ATMs, or alternately A.T.M.'s. (A.T.M.s looks weird.) chicagomanualofstyle.org, "Plurals"
The 2009 AP Stylebook's "plurals" entry has no section on acronyms, but mentions "VIPs", I can't find anything addressing how to specifically pluralize acronyms. (The "abbreviations and acronyms" section is also of no help.) [source]
The same holds for demonyms, in my opinion: NYer looks more natural (or less unnatural, anyway) than NY'er.
That said, it still seems very unusual to use demonyms in this way (LA-lino? LAXer? PDXer.. hmm... for some reason it seems more natural with airport codes), so proceed with caution.