TL/DR
Yes. This is an absolute killer for the cosmological argument, which is an incoherent argument.
The incoherence of the argument is that it conflates two different definitions of "begin to exist". One refers to a change in how matter is organized over a period of time (the everyday definition) and the other refers to matter and time coming into existence at all.
The Beginning of the Universe is a Singular Thing
The cosmological argument states: The universe had a beginning to its existence, everything that has a beginning to its existence has a cause, the universe has a cause. There are a lot of definitional problems here.
The universe is everything, and we have one example of it, and we postulate that it has existed for somewhere around 14 billion years. If by "begin to exist" we are referring to the beginning of existence itself, we have only one example of this and the cosmological argument is just an assertion about that example, and not a syllogism at all.
In order for it to be a syllogism, we have to have a definition of "has a beginning" that applies to the universe and to some other example.
This is where we have a problem.
What's a thing?
What do we mean when we say that a galaxy or a supercluster or a uranium atom or a bicycle is a "thing".
What all of those things have in common is that they are all relatively discrete bunches of stuff. "Stuff" in this context, means matter, energy, dark matter, dark energy, basic forces... the "stuff" of our universe. "Discrete" means that we can separate that bunch of stuff from another bunch of stuff by some criteria. We can call this criteria a "pattern", and say that a "thing" is a relatively discrete bunch of stuff in a pattern.
How does a thing begin?
When we say that a galaxy cluster or a bicycle or indeed any RDBS has a "beginning", what we are describing is a point in time that occurs during a transformation. So there is a big cloud of stuff in space, which is one pattern, and then over time that cloud of stuff collapses into a star, and that's a new pattern. We pick some point or some range of time during that transformation and say that the star "began" at that point.
So a "beginning" is a transformation of stuff from one pattern into another pattern. To "begin", for a star or a bicycle or a baby, requires stuff, time, and a change of pattern.
What's a Cause?
So we have stuff, and we can identify relatively discrete bunches of it, and we can see how those relatively discrete bunches of stuff can transform over time, sometimes combining, sometimes separating, sometimes merely changing from one identifiable pattern to another. This is the process that we are describing by saying that "things begin to exist".
When we observe relatively discrete bunches of stuff in patterns, we can often predict what new patterns will result from the interaction of some previous patterns. This predictability allows us to say that one thing "causes" another thing to happen. A interaction of things is a cause of a pattern if the observed pattern is a predictable result of a specific interaction.
Everything Already Exists
So we have a fairly good idea of what we mean by a thing (its some relatively discrete pattern of stuff) and beginning and cause. Notice that all of this describes a continuous process. Stuff interacts constantly, everywhere, and is always changing from one pattern to another. What it isn't doing is coming into existence and staying in existence. We don't get new bunches of stuff above the quantum scale.
And we need a prior state to say that something has a "cause", so we not only need there to be stuff, we need there to be time for the stuff to interact in prior to the beginning.
"Beginning to exist" is always a description of existing stuff being transformed. Causes always involve stuff and time. That is what these words mean. And it is what these words must mean if they are going to be used to refer to events like a statue being created by a sculptor.
The only justification that we have to assume cause and effect relationships is to use those definitions.
So did the universe begin to exist?
When we refer to the "beginning" of the universe, what we are talking about is a point which has no before. As far as we know, the universe had no preceding state before time and space existed. If it did, it's in a way that we don't currently understand and may not be capable, as a species, of understanding, any more than an amoeba can understand chess. What we know for certain is that the beginning of the universe was not the predictable result of some patterns of universe-stuff interacting, because "prior" to the zero moment, there's no stuff and no time for the stuff to interact in.
Change within the universe and the beginning of the universe are very different things. When we say that the universe "has a beginning" and something inside the universe "has a beginning" we aren't talking about the same type of event. The argument itself is incoherent because the definitions of what words mean change in the middle of the argument.
In truth, we have no justification to declare with any certainty that the universe either is an effect or has a cause. Cosmology is in part the science of figuring out whether the universe has a cause or is an effect, and the current answer is "we don't know".
A set is not a member of itself
Some specific things that have come up in comments on this and other answers. Jon Ericson, in a comment on his top ranked answer, says in a comment:
Picture the set of positive whole numbers. It has a definite start and no particular end. The number 1 is interesting because nothing comes before, but it has all the properties of all other positive whole numbers. It might have additional properties but surely not fewer. If every beginning has a cause, why is the universe different?
This, I think, illustrates exactly the problem. If we were to draw an analogy from some set to the universe, the Universe is the set of all things (in this case, the set of whole numbers). So the universe, or the beginning of the universe, isn't "1". 1 is something that exists inside the set.
Continuing the metaphor, we know that every number in the set (except 1) has a cause. What we mean by "cause" in this case is that we follow a production rule of adding one to a number, and we get a new number. So given the set containing 1 ({1}), we can see how the rules of cause and effect (add a number to a number, get a new number) produces all of the numbers in the set.
{1} is a set containing 1 and a production rule. If we existed inside the set, our entire concept of what a cause would be would be that production rule, because that's the only thing there is.
But you can't use a production rule to create {1}. By the definition used inside the set for the word "cause", {1} can't have a cause.
We're in a similar boat inside the universe. All of our concepts of cause and effect, of how basic logic works, of how thing happen... they are all production rules inside the system. We can't use the word "cause" the same way, or with the same implications, beyond the boundaries of our universe. The Kalam is simply incoherent.