That can be hard, given that the Achuar are a rather small people. I would start with the organizations of them, maybe you can find some info or contacts there: http://www.fenap.com.pe/ for Peru and http://www.territorioindigenaygobernanza.com/ecu_10.html and http://www.ecuanex.net.ec/confeniae/ for Ecuador.
You may try the Brazilian Indian Foundation, FUNAI [www.funai.gov.br] and be ready to answer why you wish to learn this language. If the people who speak it is outside Brazilian territory [see a geographic map] it is likely that they will send you elsewhere, such as the above answer. In general deep territorially encrusted small groups only have speakers situated at the very geographical local of their living área; as I read above in Equador and Peru, you may go there directly. Good luck in your endeavor. And when visiting a n indian tribe or group try to keep it clean [no smoking, for instance] since you may be looked upon as a n occidental role model. It is not uncommon to individuals to keeping your demeanor as a status symbol as well.
Try to contact the Summer Institute of Linguistics (SIL) International. There may be an Achuár Chícham New Testament translation by one of their missionaries.
I met an Achuar living in Baños Ecuador running a small souvenir stand. I got the impression from him that nowadays there are a number of urbanized Achuar. It might be possible to find one living in a city in Ecuador who would be willing to conduct online language lessons through Skype or some other video chat system.
Did you take Jose Fadul's advice above dated March 19 and contact SIL? SIL has published both a 141-page grammar of Achuar-S language (1981) and a bilingual dictionary in 1996, both by SIL linguist Gerhard Fast et al. The 141 page grammar can be downloaded for free from the SIL website. Also look at the long list of other published materials on the Achuar language on the *OLAC Resources in and about the A-S language" at http://www.language-archives.org/language/acu. Most of the 49 resources there were done by SIL. Thomas Headland, Senior Anthropology Consultant, SIL, 12-28-2015.