I've been thinking about this question for a few days and along the lines of @ThomasIngalls comment, the simple answer is that cooperation between organisms has such a huge advantage in selection that any chance a eukaryote would become a chemautotroph would simply not adapt fast enough to compete with those that are already holding these niches.
The chemautotrophs I'm aware of are usually archaeabacteria. They are optimized to grow slowly and have a large number of genes that can break down the unusual chemical compounds to grow in an inorganic often anoxic environment. These bacteria have genes evolved from the chemautotrophic earliest days of life on earth. They live deep within the earth, under ground, in volcanic vents. There are animals that live in some of these environments, but its simply too easy for these to use the bacteria that are already growing here readily rather then re-evolve the genes to do it by themselves.
Eukaryotes are already symbiotes - they used bacteria to create ATP and photosynthetically fix carbon to the extent that they absorbed them as mitochondria and chloroplasts respectively. If eukaryotes were under a tremendous amount of selective pressure, some of them might absorb symbiotic chemautotrophs, but without such pressure, it doesn't happen when you are competing against plants with roots and animals that eat for their energy.
Forming systems of organisms is much more stable than having individual species that can do it all. This question is actually similar to other questions that ask why we don't have metal armor, run as fast as cheetahs, etc. Evolution doesn't actually create super animals that are completely self contained. The ones that reproduce the best are the ones that can actually spread out the risk of survival and contribute to an ecosystem.
It kind of makes me think of human beings - the only superorganism on the planet in this regard. We don't really need to compete against any organism - we are in fact killing most of them out there without really thinking about it. That might now turn out to be good for us long term.
