(1) Everything that has a beginning of its existence has a cause of its existence.
(2) The universe has a beginning of its existence.
Therefore:
(3) The universe has a cause of its existence.
(4) If the universe has a cause of its existence then that cause is God.
Therefore:
(5) God exists.
Are there any errors in this reasoning?
Yes. The second premise is unjustified. There is nothing to suggest that the universe began to exist. The problem here is that there are a great deal of nuances in cosmology that are both difficult to comprehend and especially resistant to coherent explanation to laypersons.
This leads to several problems: sure enough, most laypersons and even some physicists misunderstand the metaphysics behind the issue; and they believe that the Big Bang Event represents the beginning of the universe's existence. Usually, though, this misunderstanding is exacerbated by statements by big-name cosmologists like Hawking that "time began to exist" in the Big Bang Event.
This is essentially "dumbing it down" for a layperson audience -- which is fine for any popularizer of science -- but it causes fundamental misunderstandings about the nature of the Big Bang Event. Time
does cease to have meaning in the familiar context if you go back far enough during the BBE; but this is only because we lack a quantum theory of gravity, and furthermore because our current models end up with finite geodesics as you trace the systemic histories "backwards."
The fact of the matter is, though, that we can't make physical assertions at this time prior to the first Planck time during the BBE. The BBE was certainly the beginning of the current
state of the universe; but there is no justification whatsoever for the assertion that it was the beginning of its ontological
existence.
For someone to make this cosmological argument, they would have to face the daunting task of justifying the second proposition. I wish them luck in this endeavor, considering that by doing so they would have surpassed the last 50 years of cosmological understanding by doing so.