"To update" is an example of an ambitransitive verb, which can be used with or without an object.
In the intransitive case, it certainly feels like the pronoun "itself" is implied, but that's not generally how this kind of phrase is considered. Consider a more commonplace, less technical example, "to shave":
- In Penny Lane, the barber shaves another customer.
- John is shaving right now.
In other languages - Spanish comes to mind - you would need a reflexive pronoun (se) in the second sentence (and in your Firefox sentence), but in English it's not necessary.
After a bit of reflection, it occurs to me that the heart of the question is "why does Merriam-Webster only list 'update' as a transitive verb?" And the answer, I think, is that the concept of a self-updating program (a self-updating anything, for that matter) is only a few years old. This is a new context for "update" - but I suspect that the next edition of MW will list "update" as both transitive and intransitive.