No, Vader says "the Rebels are here, and I believe Skywalker is with them" before the Battle of Hoth.
Also, there's a scene where the Emperor tells Vader that their new enemy is the young Rebel who destroyed the Death Star, although that doesn't necessarily imply that Vader's job is to capture him.
I actually think The Emperor is the best villain. He orchestrated the end of the Republic and hid himself right under the noses of the entire Jedi council.