The webbed feet of the Alabio duck
has neither nerves nor blood vessels, which renders the duck incapable of feeling the cold on its feet.
Subject is 'webbed feet' (plural) so verb should be 'have' not 'has'
(A) has neither nerves nor blood vessels, which renders the duck incapable of feeling the cold on its feet.
incorrect for two reasons, verb 'has' is used instead of 'have', and 'which renders' incorrectly modifies 'nerves not blood vessels'
(B) has neither nerves nor blood vessels, which renders them incapable of feeling the cold on their feet.
incorrect for the following reasons:
1. 'has' is used, correct one is 'have'
2. 'which renders' is used, if we want to refer to 'nerves and blood vessels', 'which render' should be used
3. 'them' doesn't have a clear antecedent
4. 'their' is wrong for singular 'duck'
(C) have neither nerves nor blood vessels, rendering them incapable of feeling the cold on their feet.
incorrect for 'them' and 'their' usage
(D) have neither nerves nor blood vessels, rendering the duck incapable of feeling the cold on its feet.
correct for using 'have' and 'its feet', also 'rendering the duck' is clearly modifying the entire previous clause
(E) have neither nerves nor blood vessels, rendering them incapable of feeling the cold on its feet.
incorrect for 'them' usage