I received a document where one collaborator italicized aka and another did not. Should aka be italicized?
Aka doesn't need italicising. It can be written and will be understood both with and without periods, although I see it usually written without.
Style guides will differ on the periods. As a counter-example to the somewhat heavyweight CMOS, the Guardian recommend to use aka without periods. Unusually, the guide also shows it in capitals, but a quick look in the paper shows nine lowercase aka without periods and one uppercase AKA without periods. (Edit: I asked the style guide editors and they recommend AKA because it's pronounced as individual letters.)
So: italics are not needed, and follow your own style guide (if you have one), or pick a style and apply it consistently.
This abbreviation, a.k.a., isn't derived from foreign words in the first place, so there was never a need to italicize it.
Moreover, the common abbreviations that were originally derived from foreign words, such as etc., i.e., and e.g., are considered a standard part of the English language, and most sources don't recommend italicizing those, either.
Regarding writing a.k.a. with periods, CMoS "recommends the following general guidelines in nontechnical settings" (when I would have expected to find these used in more technical or formal settings):
Use periods with abbreviations that end in a lowercase letter: p. (page), vol., e.g., i.e., etc., a.k.a., a.m., p.m., Ms., Dr., et al. (et is not an abbreviation; al. is). An exception may be made for the few academic degrees that end in a lowercase letter (e.g., DLitt, DMin); see 10.20 and rule 3.
You need some indication that it is an abbreviation rather than a word (as most answers agree). Italics usually indicate a foreign origin, so are probably inappropriate, but full stops after each letter would be fine, and capitals (AKA) would be even better. However, the abbreviation does have a flavour of police bulletins: spelling it out in full would be my preference.