Does the cosmological argument "prove that anything that exists has a cause of its existence", or is that just a premise of the argument?
There is no such thing as the cosmological argument. Rather, there is a class of arguments that share similar themes and (sometimes) logical structure, but that rely on different premises that are all referred to as cosmological arguments. For instance, a cosmological argument defended by Aquinas depends on the impossibility of an essentially ordered infinite regress. On the other hand, the Leibnizian cosmological argument depends mostly on the Principle of Sufficient Reason.
Given the wording of your question, you are most likely referring to the kalam cosmological argument, which has a premise that is similar to, but importantly different from, what you stated. The argument is usually formulated as follows:
- Whatever begins to exist has a cause.
- The universe began to exist.
- Therefore, the universe has a cause.
So, instead of "anything that exists has a cause of its existence," the premise is closer in meaning to "anything that begins to exist has a cause." The kalam cosmological argument does not even attempt to prove that "everything that begins to exist has a cause," but assumes this as a premise. This premise is often defended as self-evident or as following from a commonly accepted proposition (such as the Causal Principle), but these defenses are not part of the kalam cosmological argument per se.
If I understand your question correctly, then the answer is no. The cosmological argument takes as its premise that anything that exists must have a cause either in fieri (like me causing these words to appear by typing them) or in esse (like the words showing up on my computer screen is caused in part by the electricity supplied to it and will go away when the electricity does). The conclusion is that there must, therefore at least have been a God in order for the cosmos to exist. The fact that all things have a cause was derived by induction from our experience that such a law seems to be in place.
In fact arguments against the cosmological argument come in part directly from the fact that this is simply presumed to be true (such as Russell's counter argument which cites the Problem of Induction in that we cannot deduce that all things have a cause from the fact that all things we have experienced have a cause).