Daniel Fisher has details at Forbes. The Chamber of Commerce takes a hard line with these arrangements, but I don't see anything particularly problematic as a public-policy matter with this particular contract, which is really structured as a very high-interest contingent loan with a de facto lien on the settlement proceeds. The funder doesn't control the litigation (unless there's a secret side agreement), and there isn't a conflict of interest.
What does strike me as interesting is the existence of a "non-profit," "Friends of the Defense of the Amazon" in the Chevron litigation, which is a decidedly for-profit endeavour. I find my non-profit project decidedly restrictive in terms of what litigation I can and cannot engage in while complying with tax law, but perhaps the Lago Agrio plaintiffs have been more creative in structuring their entities.