I know that Bulma keeps calling Vegeta her husband but till now I haven't seen any proof that both of them are married. Vegeta wouldn't usually make time for such things. Are they really married or are they in a relationship?
In the manga or anime, there is no actual mention of an official ceremony where they get married. Also, considering Vegeta's attitude, he would not have the patience to endure such an ceremony.
From Dragon Ball Wiki
After approximately a year since Future Trunks' warning about the Androids, Bulma breaks up with Yamcha and becomes romantically involved with Vegeta, ultimately leading to the conception of Trunks. However, even the birth of their son was not enough to bring the two together, as of that time Vegeta prioritized his rivalry with Goku before anything else, even his relationship with Bulma. Arguments were common between the two, with Bulma often calling Vegeta a "jerk" and Vegeta showing little to no care or emotion for either his son or his future wife; this was demonstrated during an incident in which Bulma's plane is destroyed and their lives are at risk. When Future Trunks saves the two and questions Vegeta as to why he did not try to save them, Vegeta responds that he has "more important things to worry about than that foolish woman and her blasted child." Bulma later humiliates Vegeta in front of everyone when she says that every time he sticks his "Ugly Mug" up to Trunks, it makes Trunks cry, much to Krillin and Gohan's amusement and Vegeta's anger.
It is stringly hinted that the conception of Trunks was due to an one night stand between the two
Vegeta's and Bulma's indifference to each other during the Androids Saga strongly implies that the circumstances surrounding their child's conception was no more than a brief affair; Future Trunks states to Goku that in his timeline, even though Bulma fell in love with Vegeta, they did not stay together very long because their attraction was solely based on passion and loneliness. However, the seven years of peace that came after the conclusion of the Cell Games would suggest that the two settled their differences and began to have a much better relationship than previously, as they were already married by the time the Majin Buu conflict took place.
Bulma and Vegeta are married as on the Wikia for Dragon Ball Z it states she broke up with Yamcha and married Vegeta.
In Dragon Ball Z near the end with the 7-year skip, Vegeta becomes more intimate with Bulma as no problems have surfaced their world.
With the 7-year skip, Vegeta's rivalry with Goku has weakened meaning he has formed a relationship with Bulma.
it is never actually show in anime or mange but she keeps calling him her husband and trunks their bay and also future trunks justified it in his first meeting. The wiki page show Vegeta as her husband here http://dragonball.wikia.com/wiki/Bulma
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protected by Aki Tanaka Jun 1 '18 at 1:17
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