No, Billy Black is consistently portrayed as being confined in his wheel chair. Whenever he does need to go somewhere, Jacob has to drive him. You may be getting him confused with one of the other characters on the reservation, like Harry Clearwater, Seth and Leah's dad. One of the elders of the Quileute tribe like Billy, Harry Clearwater is also a good friend to Bella's father, Charlie Swan; both men enjoy fishing together. Harry has a heart attack in New Moon, and the news of his death serves as the catalyst for Edward's decision to go to the Volturi.
No Billy does not walk in the New Moon Jacob usually drives Billy around or in that book some from the pack. In Twilight when Bella first met them I think Charlie made a reference to Jacob being too young to drive but Billy responded that he had to get around so way.