It depends how immortality would work in canon for the regular Dragon Balls. Zarbon implied it wouldn't be enough to topple Freeza, but he didn't know the full capabilities of the Dragon Balls and shouldn't be treat as a reliable source.
Since it's the only real example we have, I'll bring up Garlic Jr's immortality as infinite regeneration. In this case, Vegeta eventually defeats Freeza after wearing him down, possibly getting far stronger than his canon self in this arc due to likely being able to spam Zenkais until tapping out. In a good mood, he decides to let the Earthlings go due to having achieved his wish, defeats Goku in a match after he heals just to settle the score and flies off into space. He quickly gains control of Freeza's empire, rebranding it the Saiyan Empire and finds a genetically compatible race with which to revive the Saiyan race. He remains the absolute ruler of the universe until Beerus awakes.
Meanwhile, Popo revives all of Freeza's victims and the Z-Warriors head back to Earth once they use the last wish to revive Chaozu, before heading back to Earth and reviving both Yamcha and Ten the next year. Four years later, the events of Future Trunks' timeline occur. With Trunks not existing, Future Gohan is cautious enough not to rush into battle and survives long enough to take his place in this arc. With Vegeta not showing up to fight #19, Piccolo decides to use his full power to destroy #19, with Gero fleeing at the sight and heading straight back to his lab. After that, things go relatively the same until the Rosat, in which Goku and present Gohan enter first and quickly kill Cell once they leave. Future Gohan then enters and afterwards returns to his timeline. Goku and Gohan deal with Dabura 7 years later and Boo is never revived.
When Beerus shows up, the Dragon Balls are used to transport Vegeta to Earth. With the final wish from them, they bring Tarble there after he mentions him and seeing it as the best option, Vegeta is made the Super Saiyan God. His immortality allows him to keep fighting Beerus, but is eventually defeated by Beerus sealing him away. Beerus decides to not destroy Earth as he did in canon and unseals Vegeta afterwards. He warns Vegeta that if he should ever try to usurp Beerus, he'll just be sealed away again. Not long after, the God ritual is done again to bring Goku to that level and the two start training with Whis after pleading to do so.
After this, Vegeta becomes far more recurring amongst the Earthlings due to his desire to rely on more than just immortality and surpass all due to his pride as an emperor. The Universe 6 Arc goes relatively the same, with Vegeta's immortality still not saving him from ringout. The Future Trunks Arc still happens with Future Gohan taking Trunks' role, though Vegeta has far more shine due to being able to hard counter Zamasu until the fusion. Gowasu explains Potara to Goku and Vegeta, or they find out from observation. The ToP is quite different due to Freeza not taking part. Vegeta will likely have a strong secon-in-command of his empire he's trained up or maybe an equivalent to Trunks who would take part and ultimately serve as fodder. Vegeta is likely the one to take home the win as his immortality allows him to easily defeat Jiren after Goku fights him due to Vegeta having not lost stamina. The Broly Arc never happens as even if Broly is found, Paragus' survival doesn't trigger his rage. Vegeta likely deals with Moro as well, unless Moro has some hax that even a PSSB Vegeta with immortality would struggle with.
