Filteringenabled - can't access PlayerGui?

Instead of removing the GUI from server side, delete it directly from the event in your local script.

Just because changes won't replicate to server, it doesn't mean you can't create objects or destroy them from client side. It's actually very benifitial to do so and it makes sense in this case, since UI has nothing to do with other server, it's completely local feature.

This is how it could look.

The local script:

script.Parent.MouseButton1Down:connect(function()
	game.Workspace.Events.RemoveCar.Clicked:FireServer()
	
	-- Use same loop in client script, just don't delete the car
	local found = false
	for _, obj in pairs(game.Workspace.Vehicles:GetChildren()) do
		if obj:FindFirstChild("user") then
			if obj.user.Value == game.Players.LocalPlayer.Name then
				found = true
			end
		end
	end

	local gui = game.Players.LocalPlayer.PlayerGui:FindFirstChild("Display")
	if gui then
		-- Use destroy instead of remove. Remove only sets the object's parent to nil, and might not get garbage collected if something is still accessing the object
		gui:Destroy()
	end

	if found == false then
		game.Players.LocalPlayer.PlayerGui.carRemove.TextButton.Text = "You have none!"
		wait(2)
		game.Players.LocalPlayer.PlayerGui.carRemove.TextButton.Text = "Remove Car"
	end
end)

The server script:

script.Clicked.OnServerEvent(function(player)
	-- All the server should do is destroy the car
	for _, obj in pairs(game.Workspace.Vehicles:GetChildren()) do
		if obj:FindFirstChild("user") then
			if obj.user.Value == player.Name then
				obj:Destroy()
			end
		end
	end
end)