I used to live in Santiago and Copiapo for several years when I was younger. This was during the time Rios was climbing the ranks and was becoming a world superstar -probably part of why I started playing tennis? I trained with the Campeones para Chile. Anyways, I can speak for Chile when I say that there are a lot of tennis clubs (mostly private) and I'm sure if you do a search for the names you will find the names of many clubs like: Club Alegeman (German Club) and Club Frances (French Club) and there are many others that I can't remember the name of. I know when I trained with Campeones it was at the Universidad de Catolica training site and these courts were open to the public and nicely located at the base of the mountains in the cortejeta. I never needed it, but you could even pay kids a few pesos to be ballboys while you played.
Anyways, I think if your Spanish is rusty my best advice would be to contact the Hotel at which you will be staying a few weeks in advance and talk to the concierge (they will most definitely speak English) and inform him/her of your intentions. They will be able to find out all the options for you in advance. Note that the seasons are the opposite in Chile and Argentina as they are here. Their winter is our summer and vise versa. Enjoy your trip, as these countries have so much more that red clay court tennis to offer, but if it is tennis you want to play -there is no doubt it won't be hard to find as these are very tennis orientated countries.