I don't find any evidence in the novel or the film that Tom sought approval from his adopted family. He looked at Vito as his father and Sonny as his brother. It was all straightforward. They reciprocated and Tom felt comfortable enough to differ with the Don over Sollozzo's proposal and he lambasted Sonny on a couple of occasions. On the other hand, he did not think of Connie, Mike, or Fredo as his family. He didn't think of Mama Corleone as his mother. It says so in the novel and it also says that they had been kind to him, but not loving.

There is that one scene in GFII where Tom says that he always wanted to be thought of as a brother, but I don't beleive that that statement supports any claim that he sought approval.