Well just by wikipedia-ing some South American nations, I have learned alot:
-Argentina and Paraguay had some pro-Axis elements in the army.
-Uruguay had alot of immigrants from Italy and Spain, but I am not sure if that is sufficent to push it into the axis camp.
-In Chile, Carlos Ibáñez del Campo was outsted from power in 1931, but was pro-axis.If he instead stays in power in Chile, Chile may go axis.
-In Brazil, Getúlio Vargas was pro-axis, but sided with the allies due to their control of trade.
-Peru was a democracy, and was the first nation to side with the allies, not facing a coup until 1948
-Bolivia was more moderate, and fought a war with Paraguay in 1938
-Colombia was in chaor at that time, during a psudeo civil-war known as "The Violence"
-Venezuela was unstable, but trying to become more democratic in the 30's
Now, perhaps a POD where more German, Spainish, Italian, and facist in general influence comes to South America, then some of these countries could become openly pro-axis. Probably Italian, as by 1933 and 1937, it may be to late to influence the South American countries into the axis camp. The influence could come by either trade, or even open backing of pro-facist governments by Italy. The trouble is the factions. Brazil, Argentinia, Paraguay, and Chile could all feasably go axis. Peru, Colombia, Bolivia, and Venezuela probably couldn't stand against them. However, with alot of aid from the USA, this could be done.
The truth is, there was alot of fighting between the right and the left in South America, with right wing dictatorships due to a fear of communism. Almost every nation could swing either way, into the far left or far right, although it's more likely to be the far right in most, due to control of the army. What I am saying is, almost any country could go into either camp, allies or axis. The reason they went with the allies in OTL is due to trade. Give more trade with axis, and thus more axis influence, and you may get a new front. The question is, who v. who?
Also, this research was done quickly. Feel free to yell at me and tell me I'm wrong, as I probably did somewhere.