What are they searching for in Heart of Darkness? How does ambiguity connect to a theme?

First, let's be clear about themes.  covers the basic themes of the story, but for me, the real theme is what Marlow calls the "fascination of the abomination." What this "abomination" might be is unclear -- there's your ambiguity! -- but its "fascination" is what motivates people to action. It is a kind of search for some missing, terrible secret of life. Specifically, the "abomination" might be

1. Kurtz. As a rogue agent, Kurtz...

First, let's be clear about themes.  covers the basic themes of the story, but for me, the real theme is what Marlow calls the "fascination of the abomination." What this "abomination" might be is unclear -- there's your ambiguity! -- but its "fascination" is what motivates people to action. It is a kind of search for some missing, terrible secret of life. Specifically, the "abomination" might be

1. Kurtz. As a rogue agent, Kurtz (in part) represents the undermining of capitalist authority (the Company).


2. Savagery. The world is a hostile, indifferent place, essentially at odds with civilization.


3. Greed. The abomination might be the drive to exploit the country, to extract as ivory using any means necessary.


4. Slavery. The abomination might be how the Company enslaves indigenous people, making them subhuman. 

There are many more ways you could interpret the "abomination." I think the point, however, is that this thing, perhaps best described as the mute indifference of the world towards man, is both appalling and motivating: we seek it out, symbolically, by going up the river for Kurtz, even though we know that the trip will be fruitless in the end.