Not sure what would be more natural in Esperanto, but both make sense to me. (In my native language, both "What does the fox say?" and "How does the fox say?" are perfectly correct, but in English only the first one sounds natural to me. A more experienced Esperanto speaker should clear this up.)
In "What drinks the bear?", "the bear" is the object, and "what" is the subject. So, It will be "Kio trinkas la urson?".
Many courses give sentences like the following, to make the students understand the accusative.
- Kato ŝatas muson. ... A cat likes a mouse.
- Muson ŝatas kato. ... A cat likes a mouse.
- Muso ŝatas katon. ... A mouse likes a cat.
- Katon ŝatas muso. ... A mouse likes a cat.
The same case is here. But the subject or the object is a special word called demand-vorto. :)
'How does the fox say', is natural in English. It just has to be used properly. How's the fox say hello? How would the fox go about saying hello usually or how in specific cases such as say hello to another fox already identified? How is the fox saying hello right now, for someone communicating via text or disabled such as blind.