Why am I getting the attempt to index global 'mouse' (a nil value) error?

You cannot get the player's mouse from the server which is what I think you're doing since you're using OnServerEvent.

Try doing something like this

Tool = script.Parent 

Tool.RemoteEvent.OnServerEvent:Connect(function(Player, mouseHit) --//OnServerEvent first parameter is always the Player object that fired the event
local Ammo = 5
    if Ammo > 0 then
        local Bullet = Instance.new("Part", game.Workspace)
        game.Debris:AddItem(Bullet, 2)
        Bullet.Shape = "Ball"
        Bullet.TopSurface = "Smooth"
        Bullet.BottomSurface = "Smooth"
        Bullet.BrickColor = BrickColor.new ("Dark stone grey")
        Bullet.CanCollide = false
        Bullet.CFrame = Tool.Handle.CFrame
        Bullet.CFrame = CFrame.new(Bullet.Position,mouseHit.p)

        local v = Instance.new("BodyVelocity")
        v.Parent = Bullet
        v.velocity = Bullet.CFrame.lookVector * 90
        v.MaxForce = Vector3.new(math.huge, math.huge, math.huge)
        Ammo = Ammo - 1
        print("You have" ..Ammo.. "ammo left in your gun!")
    end
end)

What I'm trying to do is when you call the RemoteEvent pass in the player's mouse.Hit.

Example:

Tool.RemoteEvent:FireServer(Player:GetMouse().Hit)

Make sure you fire the event inside of a localscript.