Noam Chomsky most often self-identifies as a libertarian socialist (which is, generally speaking, a code-word for some sort of left-anarchist), and has at other times identified himself as an anarchist. He has written (some) on anarchism, although, in all honesty, most of his writing focuses on statecraft and foreign affairs (a realm anarchists have very little direct influence on), and while his general perspective is critical of the state and state power (and most certainly of capitalism), most of his conclusions involve relying on reform (voting, protestation and appeals to rights) to an extent that many anarchists don't consider his thinking particularly anarchic. I believe he has, at times advocated for syndicalist methods of organization, but i can't cite when or where for you off the top of my head. My general take on Chomsky’s brand of anarchism is that he advocates a gradual withering away of the state, which sounds somewhat like some other lineages of thought you might be familiar with. That said, I’ve read a lot of his work and seen him speak, and I don’t think it is totally irrelevant, just not particularly anarchist.
Chomsky has also been criticized in the field of linguistics by some anarchists, though I don’t know enough about linguistics to really speak to this very well. I believe it boils down to a rather essentialist view of humans and a reliance on the idea of human nature.
On the other hand, Chomsky has often spoken against current anarchist tendencies such as more confrontational tactics and anti-civ. While these are not the be-all, end all of anarchy, he has shown precious little support for actual anarchist endeavors in the time I’ve been paying attention, unless we count publishing books and cd’s on AK Press (an anarchist publishing house that has, at times, operated in the world of anarchist publishing the way Barnes and Noble operates in the world of book sellers). This at the same time he has been relatively uncritical (and even blatantly supportive) of authoritarian lefitst Hugo Chavez in Venezuela.
Is he an “anarchist thinker or critic”? He is definitely a critic of anarchists, and he definitely does identify as an anarchist, but those things alone (for the reasons I laid out above) don’t necessarily make his thinking particularly “anarchist.”