Why are some people in South America accepting of oil, logging, and mining companies despite
their damage to the climate and their communities?

These are accepted because of the high amount of exports, contribution to GDP and growth of the country.

Explanation:

Mining, logging play a very important role in the growth and high level of GDP of South America because these are commodities of high export of the country.

Even though coal mining, reserves and logging has led to great damage to the environment in South America and affected the health of the people in an adverse way, but still it is highly practiced there because of the high amount of exports of those commodities, great contribution to GDP of the country and leading to growth.