I don't know about Peru or Bolivia, but there a lot of places to kite in Chile. October is the start of the season so you should have some good opportunities to get some.
The place with the most consistent wind is Embalse Puclaro located in the north of Chile, just east of La Serena. It's a big man made resovoir that gets thermal winds. I've never been there, but have been told that from October to April it blows 20 + knots every single day.
http://www.wannakitesurf.com/spot/South ... e_Puclaro/
Do you speak spanish? Here some links for Chilean kite websites:
(the most popular kitespots in Chile)
(local forum, post in English and I'm sure you get a response)
I don't know about gear rentals, but try this guy (Eduardo) who has a kite business in Chile and speaks English:
loadandpop <at> gmail <dot>com
Chileans are very friendly people and are generally very cordial to tourist. Post a message on the local forum and you can probably hook up with someone that can take you to the spots and show you around. Let me know if you have any specific questions.