Is it possible to save screen gui>frame>textbutton>localscript?

From your explanation, I'm assuming that you want a gui button that runs a local script after being clicked. After it is clicked for the first time, you then want the local script to run every time the player joins the game again. To do this you will need both a local script inside the gui button, and a server script in ServerScriptService

Try having this inside the local script:

function runScript()
   --enter the local script you wish to run here
end)
script.Parent.MouseButton1Click:Connect(function()
   runScript()
   game.ReplicatedStorage:WaitForChild("onButtonPress"):FireServer("")
end)
local hasPressed = game.ReplicatedStorage:WaitForChild("checkButtonPressed"):InvokeServer("")
if hasPressed then
   runScript()
   script.Parent.Visible = false --remove this if you dont want the button to go invisible if they've already pressed
end

and this in the server script:

local onButtonPress = Instance.new("RemoteEvent")
onButtonPress.Parent = game.ReplicatedStorage
onButtonPress.Name = "onButtonPress"
local checkButtonPress = Instance.new("RemoteFunction")
checkButtonPress.Parent = game.ReplicatedStorage
checkButtonPress.Name = "checkButtonPress"
local ds = game:GetService("DataStoreService"):GetDataStore("HasClicked")

onButtonPress.OnServerEvent:Connect(function(plr)
   ds:SetAsync(tostring(plr.UserId),true)
end)
local function check(plr)
   local hasPressed = ds:GetAsync(tostring(plr.UserId))
   return hasPressed
end
checkButtonPress.OnServerInvoke = check

This may have some typos, because believe it or not I'm on tablet, but hopefully it will give an outline of how it would work.