Is the ontological argument unfalsifiable?

I don't know if it can be 'falsified', but it certainly doesn't prove what it claims to, because there is no specificity in the argument. That means it offers no specific evidence that Descartes' particular God exists, and so pretty much anything can be substituted in place of the notorious G-O-D, and still be valid because it doesn't change the argument's logical structure.

Yes.  Falsifiability is a process of testing a hypothesis to fail if it can.

When you said

Of course, there is no way to empirically disprove it

I think you answered your own question, or you are seeking a distinction to redefine the term

I had to read https://plato.stanford.edu/entries/descartes-ontological/ to find out what his ontological argument was, and came away thinking he was presenting the results of an internal and personal reflection, which was not initially an argument that others should accept, but his attempt to explain why he decided God exists.  When he wrote it up, he rationalized his intuition to persuade others.

There's a lot to consider, imo, but I did not consider the question here to ask for discussion of his argument, and will conclude with the opening answer: The argument is unfalsifiable.

No, because like all arguments a priori arguments are subject to logic, and a particular a priori argument can be falsified if it can be shown to be logically fallacious.