Based on most of the game’s character designs and appearances, Batman Vengeance would seem to take place after the conclusion of Batman: The Animated Series, but before the start of The New Batman Adventures. Plot points that support this include Mr. Freeze still posing as a threat (as his appearance in The New Batman Adventures episode “Cold Comfort” was the last time he was seen prior to Batman Beyond), as well as the fact that Batgirl is the Dark Knight’s only active sidekick in the game.
Batman by this point is well established as the guardian of Gotham City, with the game sharing the voice cast, production design, and continuity of the DC animated universe. During the course of the game, Batman is monitored and assisted by Batgirl in the Batcave.
After saving the life of a woman named Mary Flynn from an apparent attempted murder at the hands of the Joker at Gotham Chemical, which resulted in the complex blowing up due to a ticking time bomb left next to a tied up Mary by the Joker, Batman investigates what possible interest his nemesis would have in the woman. He learns that Joker had abducted her son, Toby, and is holding him for a $5 million ransom and will kill both Mary and her son if she does not pay up soon, implying that Mary has high financial access. After Mary is abducted by Joker’s gang for the second time, Batman traces her location, using a tracker he previously gave to her, to a partially demolished Gotham bridge, where he comes face-to-face with the Joker once more. As the Joker seems to throw Mary’s son off the remains of the bridge, the Dark Knight saves the boy before it’s revealed to be a dummy, and Mary reveals herself to be the Joker’s companion Harley Quinn; Toby was never real and the abduction was merely a farce to lure Batman into a trap. Batman then engages the Joker and defeats him, but in the fray, the villain falls off the bridge and to his apparent death. After saving Harley from attempting suicide by jumping off the bridge, Batman lets a distraught Quinn go and monitors her activities, knowing that the Joker is most likely not dead and Harley could be his only opportunity to learn more of the Clown Prince of Crime’s true plans, making her a more valuable asset on the outside than she would be incarcerated.
As Batman continues to keep tabs over Harley, he is alerted to a crisis at a pharmaceutical company, Gotham Industrial, which is under attack by Mr. Freeze, who is angrily searching the complex for a scientist named Isaac Evers, whom he believes sent him a promotional video for a drug developed by Evers, known as Promethium, for treating conditions on the cryogenically frozen. Freeze had apparently used the drug on himself and his wife, Nora, but to no avail, infuriating Freeze with the implication that Evers had sent him the tape to taunt him. After finding Evers, Freeze abducts him and escapes the complex in a helicopter. Batman gives chase in the Batwing and eventually rescues Evers, who explains his work and what Freeze was after, lying to Batman by telling him the source of his funding originates from a grant supplied by the Wayne foundation. Knowing this to be a lie, Batman tells Evers to go into hiding, and later returns to Gotham Industrial to learn more about the source of Issac’s financial backing, eventually learning that the scientist had his funding pulled, due to Promethium being far too unstable, as it contains several flammable particles. Evers was able to successfully continue in his research after receiving funding from the Joker, with every rise in Evers’ budget coinciding with a successful bank heist of Joker’s doing, implying the villain had plans of his own for the drug. Freeze and his gang then attack the facility again to find Isaac, destroying most of Ever’s lab in the process, but are eventually stopped by the Dark Knight, with Freeze encased in ice from his own freeze gun.