In these sentences:
A: I've never seen an elephant yet.
B: I've never seen any elephant yet.
Which article fits better?
Neither article will improve the sentence. You have a clash between have never (negated experiential perfective) and yet (implying that a sighting is expected soon).
You can say I haven't seen any elephant yet in a situation where you are expecting to see one or more elephants (but it doesn't say anything about your lifetime experience). You can say I have never seen an elephant when you want to speak about your own life experience (perhaps explaining why you are excited about a potential sighting).