How do I make a text label change with local script?

As far as I understand, you are working very hard to avoid using RemoteEvents. While this might work, it is going to be a nightmare to synchronize properly. Making shenanigans like this sooner or later turn your scripts into a spaghetti code.

While your script could be fixed, I will update it to use RemoteEvent anyway. It would break for example when two players join at the same time. No more excuses, you need to start using Remotes, or you will never make a fully functional game.

Local Script:

local PayCheckEvent = game.ReplicatedStorage:WaitForChild("PayCheckEvent")

local function OnPayCheckEvent(amount)
   script.Parent.Text = "You've Received $" .. amount
    script.Sound.Playing = true
    script.Sound.Looped = false
    script.Parent.Parent.Visible = true
    wait(5)
    script.Sound.Playing = false
    script.Parent.Parent.Visible = false

end
PayCheckEvent.OnClientEvent:Connect(OnPayCheckEvent)

ServerScript:

local datastore = game:GetService("DataStoreService")
local ds1 = datastore:GetDataStore("CashSaveSystem")

local PayCheckEvent = Instance.new("RemoteEvent")
PayCheckEvent.Name = "PayCheckEvent"
PayCheckEvent.Parent = game.ReplicatedStorage



game.Players.PlayerAdded:Connect(function(plr) -- Capital 'C'
    local folder = Instance.new("Folder", plr)
    folder.Name = "leaderstats"
    local cash = Instance.new("IntValue", folder)
    cash.Name = "Money"

    cash.Value = ds1:GetAsync(plr.UserId) or 500
    ds1:SetAsync(plr.UserId, cash.Value)

    cash.Changed:connect(function()
        ds1:SetAsync(plr.UserId, cash.Value)
    end)

    local function PayCheck() -- Creating a function
        local teams = game:GetService("Teams")

        local payCheckTable = {
            ["Citizen"] = 120,
            ["Criminal"] = 150,
            ["Inmate"] = 0,
            ["Police"] = 200,
            ["HeavyGunDealer"] = 225,
            ["GunDealer"] = 165,
            ["AdminOnDuty"] = 10000     
        }

        for i, v in pairs(payCheckTable) do
            if plr.Team == teams[i] then
                return v
            end
        end
    end

    while wait(55) and plr.Parent == game.Players do --player may have left the game
        local paycheck = PayCheck() -- Triggers our function
        cash.Value = cash.Value + paycheck
	PayCheckEvent:FireClient(plr,paycheck) --fire client
    end
end)

Hope this helps. Edit: I changed player to plr in server script.