In the sentence,
"You grow taller every year." Is "grow" a linking verb or an action verb?
Yes, in this case it is a linking verb because taller is an adjective. The test is if you can substitute a different linking verb without changing the meaning of the sentence:
You are taller every year.
You seem taller every year.
then the verb in question is a linking verb.
An action verb would be: You grow string beans every year.