A great question but I fear one which does not have a conclusive answer. We only tend to focus on diseases that matter, which is predominantly human diseases are care less about other animals. Therefore there may be some animals that are more susceptible then ourselves but we have not studied them.
But with our current knowledge as it stands, humans are the animal with the most diseases.
I agree with Peter. It’s highly unlikely we have found the animal that harbours the most diseases so far. That’s because it’s really dependant on what we’ve already sampled – and we know far more about ourselves as humans than any other animal species.