Both Vegeta and Jiren went H.A.M. when the subject of the Super Dragon Balls came up during their fight; it became clear that both Vegeta and Jiren have incredibly personal reasons for wanting to get their respective wishes. We've already broken down the question of what Jiren wants with the Super Dragon Balls, so let's discuss what Vegeta may want from his wish!
The first and most obvious reason for Vegeta to want victory in the Tournament of Power may not have anything to do at all with the Super Dragon Balls. Given the type of warrior that Vegeta is, the Saiyan Prince may just want to achieve victory in order to prove that he's the strongest one in any universe!
This is basically what Vegeta is always about, whether it's his power-up rivalry with Goku, or his need to challenge and beat fighters that seemingly outclass him, like Hit or Jiren. Usually, there's pretty clear indication that Goku is the frontrunner, always a step ahead of Vegeta in terms of transformations and power-ups; however, the latest episode revealed that Vegeta may be on track for a power-up that's different from Goku's Ultra Instinct. If Vegeta achieves "Super Saiyan Blue 2" (or whatever it's called) first, then the door would be wide open for him to claim the final victory.
If Vegeta wins the tournament and unlocks a new power greater than Goku's, then he might not have that much need for a Super Dragon Ball wish. Of course, there is one frivolous desire that Vegeta may want to indulge...
Earlier in the tournament, when Universe 6 was still in the fight, Vegeta made an off-handed vow to his "student" Cabba, saying he would use his victory in the tournament to resurrect Universe 6. The promise was a callback to Cabba's own promise to Vegeta during the Tournament of Destruction, in which the young Saiyan promised to take Vegeta back to the Universe 6 version of their home Planet, Sadala, to meet the Saiyan king. Just because Dragon Ball Super hasn't made a big deal about it lately, Vegeta could still hold true to his word and fulfill his promise.
Don't think that Vegeta resurrecting Universe 6 would mean that the Saiyan Prince is getting soft or sentimental! If anything, it would be Vegeta's sense of Saiyan pride that would motivate him to resurrect Universe 6 (or at least bringing back planet Sadala in a new Universe 7 location). After seeing the potential of fighters like Cabba, Kale, and Caulifla, Vegeta could want to genuinely want to see a world of powerful Saiyan out there in the universe. It would be pretty timely, as Sadala's history is presumably the subject of the new 2018 Dragon Ball movie in the works.
Of course it's Vegeta we're talking about, so there could also be a less-than-benevolent reason for him wanting to bring back Universe 6. The more altruistic saiyans of Universe 6 aren't exactly to Vegeta's liking, and if there's a king who is forming their cultural outlook, then Vegeta may have designs on challenging his Saiyan pride in fight for leadership of their race! Imagine Vegeta going from Saiyan prince to Saiyan king of an entire planet of Saiyans he could train and command. It'd be a different (but no less exciting) kind of "power-up" for Dragon Ball Super to throw at us.
Dragon Ball Super's "Universal Survival" arc is part of the recent simulcast agreement that sites like Crunchyroll and Funimation have scored. Dragon Ball Super airs on Crunchyroll Saturday evenings at 7:15 p.m. CST. Adult Swim airs the English dub during its Toonami block Saturday evenings at 9:30 p.m, and is now available to stream on FunimationNOW and Amazon Video.