I don't know why this script for cloaking people is not working?

It would be helpful if you could give a little more information about the problem. To me, the first thing that comes to mind is that under the InputBegan event gameStuff is true. Why are you returning when gameStuff is true?

This should fix it:

wait(.5)
local plr = game.Players.LocalPlayer
local char = plr.Character or plr.CharacterAdded:Wait()
local UserInputService = game:GetService("UserInputService")
local Key = "Q"
local Visibility = 0

local function ToggleVisibility(toggle)
	-- Reverse the visibility
	-- You could also replace the if-statement with this: "Visibility = Visibility == 0 and 1 or 0" 
	-- See http://lua-users.org/wiki/TernaryOperator)
	if Visibility == 0 then
		Visibility = 1
	else
		Visibility = 0	
	end
	
	if plr.Team.Name == "Foundation Personnel" or plr.Team.Name == "Internal Security Department" then
	    wait(.8)
	    for _, Part in pairs(char:GetDescendants())do
	        if (Part:IsA("BasePart") or Part:IsA("MeshPart")) and Part.Name ~= "HumanoidRootPart" then
	            Part.Transparency = Visibility
	            char.Head.face.Transparency = Visibility
	   		end
        end 
    end  
end

UserInputService.InputBegan:Connect(function(Input,GameStuff)
    if Input.KeyCode == Enum.KeyCode[Key] then
        ToggleVisibility()
    end
end)

Note: I have also cleaned up the code a little bit. When programming you should always try to adhere to what is called the DRY (do not repeat yourself) principle. There is no need to repeat that if statement and for loop.