I recently came across an objection to the kalam argument that I haven't been able to sufficiently answer. My question is this: How can we know that the cause of the universe still exists without appealing to any sort of external arguments from natural theology that would enable us to prove this?
Thanks for this question! Technically speaking, this isn’t an objection to the “bare-bones” version of the kalam. Here’s how you can tell: ask yourself which of the two major premises (or the major and the minor, technically) does this objection undercut our warrant for or otherwise refute? Certainly not the causal principle, and not that the universe had a beginning. Thus, it’s irrelevant to the conclusion, which is that the universe had a cause.
Notice this isn’t even quibbling with Craig’s extended version of the argument, where he argues for the personal nature of the First Cause. Rather, at best what’s going on is that this is an attempt to undercut the warrant for saying that this First Cause still exists. And that is important, for if we have no good reason to think the First Cause still exists, then we have no good reason to think God still exists (inasmuch as God is identified with the First Cause).
I have a number of responses. First, why restrict our knowledge of the First Cause’s continued existence to the kalam? At first blush, one might think the objector is arguing that if other theistic arguments are imported in order to establish God’s continued existence, then it will be these arguments doing all the work, and not the kalam. And to that I say—why think a thing like that? The kalam is not about arguing God’s mode of existence, or length of existence into the future, or God’s subsequent actions. Instead, the kalam is all about the universe’s having had a cause that brought it into existence. Complaining that it doesn’t establish something further about God than it does is like complaining that the historian of the Revolutionary War doesn’t discuss when George Washington died or lived forever. It’s not, strictly speaking, relevant to the causes or characters of the Revolutionary War. Thus, the objection isn’t really objecting to much, or else is irrelevant.
Second, if we want to know whether or not God has continued existence, as that is important and relevant, then inasmuch as it is not the point of the kalam to discuss it, we’re welcome to bring in any point of knowledge that we might have. The ontological argument, the moral argument, the Bible, Christian theology, etc. all require God’s continued existence. Now an objector may protest, but so what? They don’t need to accept that God still exists in order to accept that he is the cause of the universe’s existence. Now if God as a First Cause existed, then naturalism is out the window and anything that we may know is admissible. If they want to give up naturalism and to know whether or not God still exists, we can discuss why we think it likely that if God ever existed, he still does (though I wouldn’t do this without a firm admission from them that naturalism is false—otherwise, they may just be doing this as a giant red herring). But they can’t simply say there is no reason to think God still exists if he ever did at all—at least not without argument.
Finally, in WLC’s and James Sinclair’s essay in The Blackwell Companion to Natural Theology, the extended kalam does in fact suggest that God is metaphysically necessary. This is because he is plausibly (via simplicity considerations) both uncaused as the absolute first cause and beginningless. So, as Craig says, “If the universe has a cause, then an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless, and enormously powerful . . . This, as Thomas Aquinas was wont to remark, is what everybody means by ‘God.’”
So we can see it’s not an objection to the kalam at all (it grants that God at one time existed) and so naturalism is false (or at least assumed false for the sake of argument); we are free to discuss any reasons we have for thinking if God existed at one time, he still does, including other theistic arguments, reasons for thinking that a powerful being like God is doesn’t have more powerful beings than himself (note: this need not be shown to be true—the objection is modest enough in saying that we don’t have a good reason for thinking God still exists if he ever did: so long as we have one good reason, that’s not irrational, the burden is met. It would then be up to the objector to tell us why this reason is not good for thinking that it’s probable God still exists.); and we are free to argue that an absolutely first cause that is uncaused and beginningless is very plausibly metaphysically necessary, and so we would need a reason to think that such a being has gone out of existence, which is more than what the objector has provided.
Typically, these kinds of objections are either based on misunderstandings or otherwise red herrings. If an atheist really is willing to give up naturalism (at least for the sake of argument), it’s worth it. Otherwise, it’s very likely a red herring. I hope this helped!
 Please also note that this isn’t even an argument that the First Cause does not still exist; it merely asks us how we know from the kalam.
 William Lane Craig and James D. Sinclair, The Kalam Cosmological Argument,” in The Blackwell Companion to Natural Theology, William Lane Craig and J.P. Moreland, eds. (Malden, MA: Blackwell, 2012), 194.