No. Tales of Vesperia (Xbox 360 / PS3 / Definitive Edition on Switch/PS4/Xbox One/PC) is a single-player RPG. It does not feature any cooperative or competitive multiplayer modes in any official release.
The "Party" system allows 4 characters in battle, but all are controlled by AI or a single player switching leaders (via menu or shortcut). Local co-op (multiple controllers controlling different party members in battle) exists in Tales of Symphonia (GameCube/PS2/PC), Tales of Graces f (PS3), Tales of Xillia 1/2 (PS3), Tales of Zestiria, Tales of Berseria, and Tales of Arise (online co-op added post-launch). Vesperia is the notable modern exception lacking this feature.
The Tales of Vesperia: Definitive Edition on PC has community mods (e.g., "Vesperia Co-op Mod" on NexusMods/GitHub) attempting to enable local co-op battle control. These are unofficial, often unstable, require specific versions, and disable achievements/online features. Use at own risk.
If co-op Tales is the goal, play Tales of Arise (best modern entry, online co-op), Tales of Symphonia Remastered (classic 4-player local), or Tales of Graces f (PS3/Steam, local co-op).