It is possible to have a custom animation per team?

can be can be, have Script in ServerScriptService:

local Players = game:GetService("Players")
local Teams = game:GetService("Teams")

local RUN_ANIMATIONS = {
    [Teams.Zombie] = "rbxassetid://0000000000",
    [Teams.Robot] = "rbxassetid://0000000000"
}

local function On_Character_Added(player, character)
    local run_animation = RUN_ANIMATIONS[player.Team]
    if not run_animation then return end

    -- beautiful code ain't it

    character
        :WaitForChild("Animate")
        :WaitForChild("run")
        :WaitForChild("RunAnim").AnimationId = run_animation
end

local function On_Player_Added(player)
    player.CharacterAdded:Connect(function(character)
        On_Character_Added(player, character)
    end)
end

Players.PlayerAdded:Connect(On_Player_Added)

note: this only updates the animation after respawn