Anyway to make the player's hat/part disappear?

Yes, you will, however, also need a RemoteEvent to make the changes on the server, so other players can see it.

LocalScript in Tool

local localplayer = game.Players.LocalPlayer
local character = localplayer.Character or localplayer.CharacterAdded:wait()
local TransparencyEvent = script:WaitForChild('TransparencyEvent')
local Tool = script.Parent

local WhiteList = {
	['Part1'] = true;
	['Part2'] = true
}

-- Index parts to a table, so we only need to 
	-- scan the player once
local Parts = {}

-- If Child is a BasePart and it's in the WhiteList, add to table.
local function PreCheck(Object)
	if Object:IsA("BasePart") or Object:IsA('MeshPart') then
		-- Ignore HumanoidRootPart
		if Object ~= character.PrimaryPart and WhiteList[Object.Name] then
			table.insert(Parts, Object) 
		end
	end
end

-- Recurse character when we first load
for _, Object in pairs ( character:GetDescendants() ) do 
	PreCheck(Object)
end

Tool.Equipped:Connect(function()
	TransparencyEvent:FireServer(Parts, 1)
end)

Tool.Unequipped:Connect(function()
	wait()
	-- This is so when we swap tools, it doesn't set to 0.
	if not character:FindFirstChildOfClass('Tool') then 
		TransparencyEvent:FireServer(Parts, 0)
	end
end)

-- Anything gets added to the character will be caught
character.DescendantAdded:Connect(PreCheck)

Script inside of LocalScript

local TransparencyEvent = Instance.new("RemoteEvent")
TransparencyEvent.Name = 'TransparencyEvent'

-- The Player who fired the event is always the first argument in a RemoteEvent
TransparencyEvent.OnServerEvent:Connect(function(Player, Parts, newTransparency)
	for i,v in pairs ( Parts ) do 
		v.Transparency = newTransparency
	end
end)

TransparencyEvent.Parent = script.Parent