Technically speaking, лапа is just an animal leg. So birds have лапы.
However, according to the comments, the OP is about the language feeling, so I'll elaborate in that direction. Let's compare different uses:
лапа as a gentle address to a beloved one
лапа as a reference to a very thick, unattractive leg
tigers, cats have лапы
giraffes, horses have ноги
I risk to suggest that лапа is used whenever one wants to emphasize the soft nature of a leg. Fur-covered legs give the feeling of softness and are thus called лапы. A leg is likely to be called лапа if it is thick and short, thereby imitating softness. That's why storks have no chance of having лапы, unlike, say, sparrows.