Is "update" transitive or intransitive?

According to many dictionaries including merriam-webster, update is a transitive verb. So I expect the following sentence (used in Firefox) to be incorrect:

Please wait while Firefox is updating.

I think Firefox is the object of this sentence and it should have been used like this:

Please wait while Firefox is being updated.

Is the first sentence grammatically correct?

"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.

The sentence "Please wait while Firefox is updating" is actually a condensed form of "Please wait while Firefox is updating itself"

So, to answer your question, yes, update is transitive, but modern usage has rendered some parts of the grammar as obsolete, as is evident from the sentence you gave as an example because it shortens it without actually compromising on the relevance and meaning of the sentence.