Yes, Bulma and Vegeta did sleep together and had a romantic relationship.
While the exact circumstances and details surrounding their initial intimacy aren't explicitly depicted in the Dragon Ball series, it's clear they became romantically involved during the Android Saga. This resulted in them having a son, Trunks.
Different versions of the anime and manga have offered varying nuances to their relationship:
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Japanese Version: Some interpretations suggest a more casual initial encounter, possibly akin to a one-night stand, based on implications and dialogue.
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American Dub: The American dub often portrays their relationship as a shorter, but more explicitly romantic, affair.
Regardless of the initial circumstances, their relationship developed into a long-term commitment. Vegeta eventually admitted his love for Bulma, solidifying their bond, even if it was initially triggered by Majin influence which amplifies hidden feelings. Their continued interactions and subsequent child, Bulla/Bra, clearly show a deepening connection.
