The argument runs like this:
(1) Everything that exists contingently has a reason for its existence.
(2) The universe exists contingently.
(3) The universe has a reason for its existence.
(4) If the universe has a reason for its existence then that reason is God.
(5) God exists.
I have two immediate objections to the argument, which I'll call the Nature of Necessity objection and the Quantum Mechanics objection. I've already discussed QM and reasons in my debate with Prof. Feser (see here and here), so in this post I'll just address the Nature of Necessity objection.
Cosmological arguments always begin by implicitly carving out a "God exception."
- - "Nothing can be beginningless - except God."
- - "Nothing can exist without a cause - except God."
- - "Nothing can be undesigned - except God."
Properly speaking, "necessity" and "contingency" are terms that apply to propositions. Necessary truths are those that cannot fail to be true: the usual examples are mathematical and logical truths. Contingent truths could fail to be true. "It is raining right now" might be true, but it could have been the case that it didn't rain today.
The argument from contingency rips these terms out of their proper context and attempts to apply them to things. It asserts that there are contingent things - those that could possibly have not existed - and necessary things - those that could not have not existed. The problem is, no one ever seems to be able to point to something that exists necessarily - except God.
"Exists necessarily" is supposed to mean "exists in all possible worlds." But what is a possible world? Philosophers distinguish different types of possibility. Physical possibility means that which, according to the laws of nature, could happen. Logical possibility means anything that is not logically incoherent. (There are other types of possibility, but these are enough for discussion.) In making a modal argument - one based on what might happen in other "possible worlds" - it makes a big difference whether we are talking about physically possible worlds or logically possible worlds. So, it might be logically possible for the universe not to have existed, while at the same time it is physically impossible for the universe not to have existed. (That would be the case if the laws of physics somehow guaranteed that a universe spring into existence. I'm not claiming that they do guarantee this - only saying that some set of laws might do so.)
The POR article considers logically possible worlds and suggests that it seems logically possible that the universe not exist. This is, of course, true. Among the logically possible worlds is the empty world, in which nothing exists. So, no universe.
But also, no God. Because when necessary existence is construed according to logical possibility, it is an empty concept. Nothing exists in the empty world. So there is nothing that exists in all possible worlds. So there is nothing that exists necessarily.
The Nature of Necessity objection was articulated by Bertrand Russell a long time ago, so I'm sure someone has thought up a response to it. If you know of one, let me know!