Is the word disputar the same in Brazilian and European Portuguese?

Yes.

The word "disputar" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only differences are phonological. In Brazilian Portuguese, the vowels tend to be more open, and the final "r" is often aspirated (sounding like an "h") or silent, depending on the regional accent. In Continental Portuguese, the vowels are more closed or reduced, and the final "r" is typically more clearly articulated as a tap or a subtle trill.

Brazilian Portuguese

• O time está disputando o campeonato. (The team is competing for the championship.)
• Eles estão disputando o brinquedo. (They are fighting over the toy.)
• Os candidatos vão disputar a vaga. (The candidates are going to compete for the vacancy.)
• Os jogadores estão disputando a posse de bola. (The players are competing for possession of the ball.)
• Eles vão disputar o prêmio. (They are going to compete for the prize.)

Portuguese (Portugal)

• A equipa está a disputar o campeonato. (The team is competing for the championship.)
• Eles estão a disputar o brinquedo. (They are fighting over the toy.)
• Os candidatos vão disputar a vaga. (The candidates are going to compete for the vacancy.)
• Os jogadores estão a disputar a posse de bola. (The players are competing for possession of the ball.)
• Eles vão disputar o prémio. (They are going to compete for the prize.)