Why can't I see the LazerShot when my character is in first person?

Its cause your parenting it to the character. When you go in first person. Everything in the character becomes completely transparent. I'd recommend adding it to the workspace. Maybe the tool. But parenting it to the tool's parent parents it to the character. and that just won't work.

What i would personally recommend is using game:getService("Debris")

So i'll try editing your script to use "Debris". Just minor differences

Tool = script.Parent
Handle = Tool:WaitForChild("Handle")
SoundModule = require(game.ReplicatedStorage.SoundModule)
Damage = require(game.ReplicatedStorage.DamageModule)


local Debris = game:GetService("Debris") ---Define The Service "Debris"
Players = game:GetService("Players")

Speed = 100
Duration = 1

ServerControl = Tool.ServerControl
ClientControl = Tool.ClientControl

NozzleOffset = Vector3.new(0, 0, -15)
function Equipped()
    Character = Tool.Parent
    Player = Players:GetPlayerFromCharacter(Character)
    Humanoid = Character.Humanoid

    if Humanoid.Health == 0 then
        return
    end
end
Tool.ServerControl.OnServerEvent:Connect(function(player, Mode, Value, arg)
    if Mode == "Click" and Value then
        Activated(arg)
        print("Fired")

    end
end)



function Activated(target)
    if Tool.Enabled then
        Tool.Enabled = false
        local HandleCFrame = Handle.CFrame
        local FiringPoint = HandleCFrame.p + HandleCFrame:vectorToWorldSpace(NozzleOffset)
        local ShotCFrame = CFrame.new(FiringPoint, target)

        local LazerShot = BaseShot:Clone()
        Damage.Damage(LazerShot,Tool.Parent.Humanoid, 14, 1)
        LazerShot.CFrame = ShotCFrame + (ShotCFrame.LookVector * (BaseShot.Size.Z/2))


    --local BodyVelocity = Instance.new("BodyVelocity")
        --BodyVelocity.velocity = ShotCFrame.lookVector * Speed
        --BodyVelocity.Parent = LazerShot
	Debris:AddItem(LazerShot, 10 -- The time you want the lazershot to bed active/not destroyed, So after 10 seconds in this script it will be destroyed)
        LazerShot.Parent = workspace -- So it doesnt disappear when you zoom in
        SoundModule.Sound(Tool.Handle.Fire, Tool.Parent.Torso, .6, 0)
        wait(.5)
        spawn(function()
            for i = 0, 1, 0.08 do
                wait()
                LazerShot.Transparency = i
            end
            LazerShot:Destroy()
        end)
        Tool.Enabled = true
    end
end

BaseShot = Instance.new("Part")
BaseShot.Anchored = true
BaseShot.Name = "Effect"
BaseShot.BrickColor = BrickColor.new("Bright green")
BaseShot.Material = Enum.Material.Neon
BaseShot.Shape = Enum.PartType.Block
BaseShot.TopSurface = Enum.SurfaceType.Smooth
BaseShot.BottomSurface = Enum.SurfaceType.Smooth
BaseShot.FormFactor = Enum.FormFactor.Custom
BaseShot.Size = Vector3.new(0.5, 0.5, 55)
BaseShot.CanCollide = false