Whenever I Run A For Loop It Determines Instance Isn't Nil.. Even When I Know It's Nil .. Help?

It's a logic error. If you notice on line 6 (you do the same on line 9)

    if v.Backpack:FindFirstChild(WeaponToFind) == nil or v.Character:FindFirstChild(WeaponToFind) == nil then

What is happening is that the weapon wasn't found in the character so v.Character:FindFirstChild(WeaponToFind) == nil was met, or it was in the character and v.Backpack:FindFirstChild(WeaponToFind) == nil was met. When you use or in a conditional statement, at least one of the conditions have to be truthy (not false or nil), and since Instances are not false or nil, your condition is met.

You want to switch that 'or' to 'and' so that BOTH must be met (no weapon in backpack AND character).

local playing = game.ServerStorage:WaitForChild("GameState"):WaitForChild("Playing")

playing.Changed:Connect(function(isPlaying) -- much better than a while loop
    if isPlaying then -- if the new value is true
        for _, player in ipairs(game.Players:GetPlayers()) do
            local WeaponToFind = player.Stats.LoadoutGun.Value
            if not player.Backpack:FindFirstChild(WeaponToFind) and not v.Character:FindFirstChild(WeaponToFind) then
                local WeaponToGive = game.ReplicatedStorage.Weapons:FindFirstChild(WeaponToFind):Clone()
                WeaponToGive.Parent = player.Backpack
            end
        end
    end
end)

I also used a Changed event rather than a while loop. You only need to check when it changed, not every 0.5 seconds. And you don't need that elseif, and the == nil was not needed, because as I said, not nil would be true, since 'nil' and 'false' is falsey, and :FindFirstChild() returns nil if there isn't a child found.

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