How do I make it so a button would change the Local Player's gui interface?

So, for this, I would suggest adding a remote event.

The general layout should look like this.

The click script should be the following:

script.Parent.ClickDetector.MouseClick:Connect(function(plr) -- On clicked
    game.ReplicatedStorage.ShowGui:FireAllClients() -- Fires to all clients
end)

This notifies every client that the button has been pressed.

The LocalScript should be the following:

game.ReplicatedStorage.ShowGui.OnClientEvent:Connect(function()
    -- WHATEVER YOU WANT TO HAPPEN WHEN BUTTON IS CLICKED HERE
    script.Parent.Enabled = true
    wait(4)
    script.Parent.Enabled = false
end)

You could try a bindable event