Does determinism work as a viable criticism of Descartes' 'I think, therefore I am'? If determinism says that our thoughts are a product of a a biological and chemical system, over which the thinker has no control, does that go as far to say that the thinker is not him/herself thinking, but passively being given thoughts and actions?

I understand that quantum theory means that true determinism is not possible. But this does not alter the lack of free will, or having active choice over our thoughts, that determinsm presents.

No. It could just as well be stated "I think, therefore a thinker exists." Descartes was wrong, however, in his conclusion for another reason. He didn't define the thinker, "I", so his statement implies that a homunculus exists apart from the thinking going on. The more nearly proper statement would be "I think, therefore thinking exists."

Whether the thinking process is deterministic or not, it is occurring, so we can imagine an intelligent robot, fully deterministic, saying something similar to Descartes' famous remark.

That still leaves the problem of defining what thinking is, and leaves a further (and harder) problem of consciousness. Some thinking is conscious, some isn't.