Does he make the claim that thinking cannot be rationally explicated to any extent, ie given some kind of structure?
It is important to understand with Wittgenstein that he has a particular understanding of 'explanation' which differs from that most commonly found in analytic philosophy. Commonly, 'explanations' are given by formal analysis - most typically, in recent decades, giving the structure of sentences or propositions in terms of a calculus (this holds true of logical positivism and truth conditional semantics, the two main strands of linguistic analysis in the 20th century).
Wittgenstein, by contrast, sees such activities as providing not inner structures but representations of language - he stresses that there is no single true representation and nor need there be. A 'proposition' is a 'family resemblance' concept and has no single structure or essence. Explanation of language, for Wittgenstein, is given through showing the underlying grammar (which differs from the superficial or 'surface grammar') of sentences. Grammar, in this sense, is determined by how the sentence is really used (not only by philosophers!), and what connections it has with other sentences.
It would be mistaken, on such an account, to think there is a grammar of thinking. If one wishes to understand what thinking is, one could look to the uses of the word 'thinking' and its cognates. Philosophy has no other means of explaining a term. If there is some other discipline, a science or empricial psychology, which can explicate thinking and any 'structure' it may have, then that is fine, but not the domain of Wittgenstein's investigations and not the domain of philosophy.