If you are talking about Vegeta as a kid, I have seen him fight in the movie-Bardock-the Father of Goku (if i remember correctly).
If you are talking about Vegeta Jr. (the great-grandson of vegeta), he is seen fighting Goku Jr. in the last episode of Draghon Ball GT.
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if I'm not mistaken its episode 280 in the uncut version and the episode is called "Vegeta's Respect"
well goku fights in an episode called "Battle for the Universe Begins" in the uncut series it's episode 279 that they start in and if I'm not mistaken its episode 280 in the uncut version were vegeta fights buu and the episode is called "Vegeta's Respect".
episode 213
Vegeta turns ssj4 in episode 59 of dragon-ball gt
episode 214 when hes majin