I'd like to obtain residency through my newborn son here. Or alternatively through my Argentinian wife, whichever is easier. I don't have my US birth certificate, so I think I have to fly back to the U.S. and order a certified copy of one, apostilled. Based on my google research, that appears to be the only document I need from the U.S., other than my passport?
I don't know what state you were born in, but I believe it's highly likely that you will be able to order a certified copy of your birth certificate on line from the office of the Secretary of State in which you were born and they should be able to perform the Apostille
for Argentina as well.
I was born in Illinois did this on line from Argentina in 2006. I think I had to have it sent to a US address and that was thee case I had it sent to a friend in Chicago who then sent it to me (along with some other docs including my US criminal report).
If you apply for permanent residency in Argentina you may need a US criminal report from the FBI as well. Especially if you lived in the USA for even one year out of the last ten...and it appears (based on a recent post by another member) that means 365 days out of the last ten years combined!
As you have a son who was born in Argentina you "should" be eligible for Argentine citizenship straight away. It costs next to nothing compared to getting permanent residency and you shouldn't nee a lawyer for that, either!
Just go to the federal court that has jurisdiction over your domicile and ask what they need! At some point they may ask for an FBI report but not up front.
PS: And let your wife do the talking....
PS2: Congratulations to both of you on the birth of your son!