The question "Can I work with a student visa?" has been asked several times. I believe the answer has always been yes, but I don't remember if anyone posted whether or not the employer must be registered with migraciones to hire foreigners.
Cerati posted that "The employer is not going to investigate how you have you DNI or residency as long as the DNI is not expired." but isn't there still a risk to the employer if AFIP investigates?
It is my understanding that the employer
does not need to be registered with Migraciones / "eligible" to hire foreigners to employ a foreigner who has temporary residency as a student. You just need a CUIL and a DNI so they can register you with AFIP and make the necessary tax contributions (aportes) in your name. Migraciones and AFIP are pretty separate; however, temporary CUILs may eventually expire in the system and cause problems, I don't know.
However, if an employer wants to petition on your behalf (via a pre-contract to be presented at Migraciones) for your application for temporary residency as a "trabajador migrante" they
will have to be registered with the ReNURE of the Dirección Nacional de Migraciones. (Just as any educational, degree-granting institution who wants to "sponsor" foreigners for residencies as students. I had an issue once renewing my temporary residency because the UBA hadn't renewed their registration yet.) This requires being in good standing with AFIP, having your bookkeeping and actas in order, as you have to present the most recent documents to register. So it's maybe not something a business that does a lot of off-the-books stuff would want to bother with (restaurants, for example, hostels, or language institutes).
Basically I think the issue is that most small businesses are not really interested in taking this trámite on because it's not worth it to them. And medium/large businesses, unless they work with a lot of non-Mercosur foreigners because they are a multinational or the local branch/subsidiary of a US or European corporation, aren't educated on this issue and would rather not take on the hassle of petitioning on your behalf. They just see a foreigner and think, ugh, paperwork, what's in it for me??
Anyway, here are the requirements for getting residency as a "trabajador migrante" here:
http://www.migraciones.gov.ar/accesible/indexN.php?nomercosur_temporaria (Click on the section entitled "Trabajador Migrante")
In my opinion the website is fairly straighforward, but I guess it's a matter of opinion. If you have a question regarding specific details of your immigration status/application, unfortunately, like many things, it's best to go and inquire personally! A pain in the neck, but if you factor into your timeline, at least you don't have to rush.