Speak with the woman in the lobby at the Symposium on the Future of Man to get your very own robotic fighting monkey, Mr. Tails.
The first Robot Monkey Fighting card is already in your possession at the start of the game, having been received from Dr. Ernest Whimple as a reward for completing his tasks in the first game. When you first enter the Robot Monkey Fighting arena it will be placed into one of the slots on Mr. Tails' back - although he doesn't perform as well as you might hope.
The second is given to you by Dr. Crazoir when you get his release papers during Case 3: The Manacled Machinist. However, even with this card inserted, Mr. Tails doesn't emerge victorious from a second attempt in the ring.
The third Robot Monkey Fighting card is found in Wolfington's apartment during Case 1: The Housing Crisis. With all three cards installed in Mr. Tails, he's more than a match for his opponent and will handily defeat the larger monkey robot.
As a reward for closing this case, you'll receive the Robot Monkey Fighting League trophy, which contains the Energite molecule needed for Case 5: What A Girl Wants.