The Saiyan Saga is one of the biggest arcs in Dragon Ball history. It establishes Goku's Saiyan origins, implies the existence of a planet-trading organization, kills off several important characters, and instigates the trip to Planet Namek, thus leading to the events of the Namek Saga and everything afterward. One of its biggest influences, however, is the introduction of Goku's eternal rival Vegeta.
Another important thing about Vegeta is that he's Bulma's current love interest. They seem to live together happily and even have two children. With that in mind, maybe the Saiyan Saga, despite its hardships, isn't such a bad thing. In fact, it's entirely possible that Bulma indirectly wished for Vegeta to come to Earth. Here's a theory about how Bulma may have used the Dragon Balls to wish for Vegeta.
According to one Redditor's headcanon, Bulma could have used the time between Dragon Ball and Dragon Ball Z to gather the Dragon Balls and summon Shenron. Afterward, she'd ask for a boyfriend better than Yamcha. Rather than grant this wish directly, however, Shenron would manipulate events so that Vegeta would find his way to Earth and, eventually, to Bulma.
The theory isn't an unfounded one and there is some logic behind it. There are five whole years between Dragon Ball and Dragon Ball Z, leaving plenty of time for anyone to make a wish. The wish references her original purpose for gathering the Dragon Balls; she was originally going to wish for a boyfriend but ended up meeting Yamcha, so she didn't need the Balls anymore. Bulma ends up leaving Yamcha for Vegeta because she gets fed up with him playing the field. When she's first seen at the start of DBZ, she's so mad at Yamcha's that she doesn't even tell him that she's meeting up with the others. With all of this in mind, it's entirely believable that Bulma might have used the Dragon Balls to impulsively wish for a new, better boyfriend.
The commenters have some pretty interesting responses to this post. Some bring up how the timeline doesn't line up since Goku uses the Four-Star Dragon Ball for Gohan's hat (and has a couple more Balls at home). Some doubt Shenron's ability to work like a monkey's paw; the Eternal Dragon should only be able to grant wishes directly and as intended. One commenter suggests that if Bulma did this, she would have wished for a prince who was strong like Goku but also smarter. One commenter brings up how they want a villain who would use the Dragon Balls to wish themselves into the strongest warrior in the universe; this is off-topic, but it's also precisely what happens with Gas in Dragon Ball Super five years after this comment was posted. There are all sorts of comments with varying levels of agreement and disagreement though things become clearer with some further examination.
Despite the initial sense the theory makes, it's unlikely that events would have played out like this. For one thing, however capable Shenron is of granting this wish, it seems like manipulating fate would be well beyond his power. It's also unlikely that Bulma would go out of her way to scour the globe for all seven Dragon Balls to make such an impulsive wish. The more one thinks about it, the more holes this theory has.
The biggest case against this theory, however, is Bulma's genuine love for Yamcha at this time. Despite all their bickering, Bulma is always willing to let Yamcha come back to her. When the Turtle Hermit comments on her anger at Yamcha at the start of DBZ, it implies that their arguments are simply par for the course. This love is also shown in her reaction to Yamcha's death. When this happens, she falls into an unfathomable sadness that can't be fully comprehended by even her closest friends. She could only feel this way if she truly loved Yamcha.
Whatever the truth behind Bulma leaving Yamcha for Vegeta, this is a fun theory to postulate about. The idea of Bulma literally wishing for a new boyfriend seems in character enough for her. Depending on how fans view its validity, it can make for an excellent headcanon.
