You need to acquire a Chilean tax ID before buying a car, but that takes only a few minutes. It's better to enter Argentina with the official
padrón (
tarjeta verde to Argentines), which takes a little while to arrive, but you can do so with the bill of sale. As I understand it, though, an Argentine citizen or resident
cannot take a Chilean vehicle into Argentina. In fact, an Argentine citizen
cannot even drive that foreign vehicle, so when my wife and I travel together on the Argentine side I have to do all the driving.
I've covered this topic in more detail at
http://tinyurl.com/pbnjoa2