Is there still a need to trade with developing countries?

Yes, there is...for 2 main reasons.

1) The example you give (of cheap labour moving to the US) is not particularly good because as soon as that cheap labour moves to the US, it will become more expensive. This is because those workers will be paid under the minimum wage system in the US, and so the effect that cheap labour has on the US economy is minimised. Therefore the labour found in countries like China will be much cheaper than in the US for many years into the future.

2) You assume that the only thing that developing countries have to offer is their cheap labour, whereas many of them export important agricultural commodities (e.g. palm oil, sugar etc.) and raw materials (oil, metals etc.).

Therefore, not only do these countries offer the cheapest labour (as long as those workers stay there), but they offer products that some developed countries have no opportunity to create.

(1) It isn't a question of "need"; it is a question of want because we as an economy are better off if we do,

(2) You are right to note that if the U.S. allowed more unskilled, low-wage, labor into the country, there would be less incentive to outsource production to developing countries
http://www.nytimes.com/2010/10/31/business/economy/31view.html

(3) On the other hand, the key idea of classical trade theory is that _every_ country has a comparative advantage in something, and this includes all the developing countries.
http://www.econlib.org/library/Topics/Details/comparativeadvantage.html
http://tutor2u.net/economics/content/topics/trade/comparative_advantage.htm
http://www.flatworldknowledge.com/pub/international-trade-theory-and/199668#web-199670

So even with greater immigration, there are some products in which they will have a comparative advantage and the U.S. will want (not need) to buy those goods from them.