Natural Law is sometimes said to be the product of "God, nature, or reason." Even if you focus on God, it is immediately apparent that there has been a broad consensus with regard to theories of Natural Law, even among people with very different conceptions of God, from Aristotle to Thomas Jefferson.
One way to look at this question is to turn it round and look at the alternative to Natural Law Theory. If there is no Natural Law, then all law is what is known in legal philosophy as Positive Law, the body of legislation enacted and enforced by the state. In The Science of Ethics, Sir Leslie Stephen suggests an extreme thought experiment in which the British parliament enacts a statute demanding that all blue-eyed babies should be killed. In this case, according to Positive Law "the preservation of blue-eyed babies would be illegal." However, Stephen points out, no sane legislature would enact such a law, and no intelligent subjects would obey it. This is because the law is obviously immoral. To raise such an objection, however, you must believe that law should be based on morality, which is to say that you believe in some kind of Natural Law Theory.
The question of whether there can be a reasonable Natural Law Theory without a theistic basis, therefore, is essentially the same as the question of whether it is possible to have morality without God. While some religious people claim that all morality is based on divine command, in practice the divine commands on which they rely in contemporary society are generally the same as those which atheists and agnostics reach through a combination of reason and empathy. Where this has not historically been the case, Western liberal democracies have tended to decide that laws based solely on religious principles should no longer apply, as with, for instance, the criminalization of homosexuality. If you examine the Positive Law of the United States, or or any Western European state, you will find that it now conforms more closely to the principles laid out by John Stuart Mill in Utilitarianism than to the commandments in the Bible. Utilitarian philosophy is one of many reason-based theories of justice which do not require any theistic basis.