I'll offer that the information you seek is certainly available on the HUBB as well as scattered all over the internet. It might just be that you don't find it all neatly and coherently assembled in one place. What would that look like, anyway? Something which describes a family just like yours doing just the trip you envision within the past couple of years? Unlikely. Instead, you'll find it in pieces and you'll assemble the whole picture bit by bit.

In practical terms, asking specific questions will tend to yield useful answers here and elsewhere. I'll address just one question which you are not exactly asking, but which seems to me key: your budget. You're looking at about 54 euros per day, which is pretty low for three people in a 4x4 camper van. In fact, it's lower than average (although within a reasonable range) for one person on a motorcycle. If that budget includes, for example, flights to and from South America, vehicle purchase with insurance and registration, any necessary gearing-up or vaccinations, or periodic splurges on food, lodging, souvenirs, tours, etc., you're going to want to reconsider.

Chile's one place where you can legally purchase, subject to all sorts of caveats. Lots of people go the quasi-legal route in various countries. This always involves photoshop at some point, and it's probably the norm for people in your position, both purchasers and sellers. Lots do ship vehicles from Europe, Central America, or North America--you could start by perusing the shipping section here on the HUBB.

There is no place where it is inexpensive to purchase a vehicle--at least by my North American standards.

Hope that's helpful.

Mark