There is no official way to get the Batmobile in Grand Theft Auto: Vice City on PS2. The Batmobile is a DC Comics copyrighted vehicle. Rockstar Games never licensed it for Vice City.
Modding PS2 games requires a modded console (FreeMCBoot, modchip) and patched ISO files. Risks: disc read errors, save corruption, hardware strain, legal issues. PC is the only viable platform for GTA Vice City mods.
Enjoy Vice City's actual 1980s car roster—it's one of the best in the series!