So I can answer this a bit plainly here.
It's a little yes and a whole lot more no.
The reasoning behind this is purely in terms of American pressure and the fact of the matter most if not all of South America relied heavily on US goods for many purposes. Albeit I'll point out under Getulio Vargas there was a lot of push to more actively industrialize Brazil. This however does not make up for the disparity in industrial capability between USA and pretty much the entire South American continent. There simply isn't a comparison to be made here. The USA was so utterly superior in industrial capability that it would be no contest even with the USA being unprepared conventionally against countries like Germany (the US army was more or less operating on an extremely low tier until the first drafts brought the army into respected size and capability)
Given the disparity that didn't stop Germany from invading pretty much all of Europe! Which I've heard as the big retort against this. Well first of all Germany was still an industrial giant whereas nothing in South America was even close to performing or manufacturing the same amount of equipment the Germans did. There simply isn't enough industry or even manpower (Brazil the only thing that could be considered a competitor under the most ideal conditions) had 40 someodd million people to the USAs 120 million). To me this definitely seems like a recipe for disaster if they joined the Axis. Regardless of political leanings the South American counties simply lacked pretty much everything they needed to fight war on that scale. Before this the largest war south America had ever seen was the Paraguayan war which was devastating but pretty much every single side in that conflict performed absolutely poorly. So without a big military tradition, industrial capability, manpower, or equipment everyone wisely pulled together that it would be suicide to fight the Allies specifically since joining the Axis would almost certainly look like a violation of the Monroe doctrine to the Americans. If anything it would probably force America into the war earlier!
The real meat of it though is it simply would be so much more beneficial for the South American countries to profit off of helping the United States and the Allies in the long run. It also helps that special council's one notable being from the Rockefeller family were appointed to help Latin America and give them clear incentives not to join and remain neutral. Considering Mexico sent volunteers and Brazil properly joined the war and sent a few divisions to italy to assist
Leonard, Thomas M.; John F. Bratzel (2007). Latin America during World War II. Rowman & Littlefield. ISBN 0742537412.
Morris 1960, pp. 129–135