Can I do Instance.new through a local script?

There's a difference between using LocalScript and Script. If you want other people see the new part created, you should make in from the server. You can use service Players and use the event called PlayerAdded for when a player joins and from there use the event CharacterAdded from the player object for when the character is added and it works if you want to do something every time the character loads. There's a example to help u understand

--This needs to be done in a server script, not a localscript

local Players = game:GetService("Players") --Here we get the service Players to use its events


--Here we connect the event for when a player joins. The function that we connected
--Receives one argument that we can call whatever we want, that argument corresponds to the player that joined
Players.PlayerAdded:Connect(function(Player)

    --Next we do is connect the event CharacterAdded from the PlayerObject
    --This events fires every time the player resets or dies.
    --We connected it to a function that receives one argument, in this case the Character
    Player.CharacterAdded:Connect(function(Character)
        --Here we can Instance the part and do whatever we want with the character

        local shirt = Instance.new("Part")

        --Here's your code, in this case we need to replace this, since we already have the Character.
        --shirt.Parent = game.Players.LocalPlayer.Character.HumanoidRootPart

        shirt.Parent = Character.HumanoidRootPart

        --shirt.Anchored = true

        shirt.CanCollide = false
        shirt.Size = Vector3.new(2.32, 2.32, 1.32)
        shirt.Transparency = 0.35
        shirt.Name = "Anti-Skid Shirt"
        shirt.TopSurface = "Smooth"
        shirt.RightSurface = "Smooth"
        shirt.LeftSurface = "Smooth"
        shirt.BottomSurface = "Smooth"

        --[[game:service'RunService'.Heartbeat:Connect(function()
            shirt.CFrame = game.Players.LocalPlayer.Character.HumanoidRootPart.CFrame
        end)
        This piece of code can be improved, you can weld the part to the HumanoidRootPart instead of changing the CFrame every frame
        We'll use WeldConstraint, the part that we created need to be unanchored, that's why i didn't anchor it
        ]]

        shirt.CFrame = Character.HumanoidRootPart.CFrame --We need to move the part to the HumanoidRootPart CFrame so it stays in that place

        local WeldConstraint = Instance.new("WeldConstraint", shirt)
        WeldConstraint.Part0 = shirt
        WeldConstraint.Part1 = Character.HumanoidRootPart


    end)


end)