I'd better point out first that this absolutely misrepresents West: he does not even mention theatre. To claim that he is making any kind of argument about tragedy is an outright lie.

Even aside from that it's hard to decode exactly what the person who wrote this line meant, because it is so badly written. What is "early studies in Greek religion and theater... was heavily influenced by Central Asian shamanistic practices" supposed to mean? I don't know. Did they mean to write "early Greek religion was heavily influenced by Central Asian shamanistic practices"? That's my best guess. It's still wrong, and it still misrepresents West's argument.

{Edit: it would appear I was right. The phrase "early Greek religion was heavily influenced by Central Asian shamanistic practices" does appear in this Wikipedia article.}

The actual source of this claim appears to be Eli Rozik's 2002 book The Roots of Theatre: Rethinking Ritual and Other Theories of Origin, which was in turn based on two books (1970 and 1975) by E. T. Kirby. Kirby's theory must in turn be based on a misreading of Dodds' 1951 book The Greeks and the Irrational, which covers the subjects of "shamanistic" influences on Greek religion (pp. 135-78) and Euripidean tragedy (pp. 179-206), but in separate chapters, and without any claim of linking the two. Looking through the history of the Wikipedia article, I can see that Rozik was once cited there as a source alongside West's book, but that citation was removed in September 2013, thereby making the paragraph entirely false.

That's not to say that West's actual argument is robust, mind. What he actually claims is that (1) Central Asia and Siberia are where "shamanism" finds its "fullest expression"; (2) a story of Dionysus in a lost Hellenistic poem, known as the Orphic Rhapsodies, shows elements from two types of "initatory death" which he associates with this monolithic "shamanism"; (3) there are potential avenues for influence, since Greeks of the time had close links to Thrace and Scythia/Ukraine, and those regions are linked to some figures in early Greek religion that have sometimes been compared to shamans, namely Aristeas of Proconnesus, Hermotimus of Clazomenae, and Pythagoras.

West's description of the contents of the Rhapsodies is solid enough, but steps 1 and 3 in the argument are not. There is no such thing as a monolithic "shamanism": that is only an umbrella term for a bunch of characteristics of many Asian religions that have been violently lumped together by people like Mircea Eliade. Step 3 is largely pointless because the steppes of Ukraine are not in Siberia; additionally, in the case of Pythagoras at least, the supposed links to Scythia were a very late invention, not attested until the 2nd century CE (Clement of Alexandria, followed by Iamblichus in the 3rd-4th centuries).

The main reason we're talking about "shamanism" in Greek mystery religions at all is because of Dodds' book. Thanks to Dodds and Burkert, it was for quite a while customary to describe the hierophants or magoi in Greek mystery religions, along with the figures West mentions, as shamans. That practice has come under heavy fire especially in view of work on the magoi described in a 5th-century-BCE tract called the Derveni papyrus, where it's clear that the magoi are not Persians but specifically Greek hierophants.

Oddly enough, though, there could be some basis for linking these religious figures (the magoi) to tragedy; it's just that Rozik doesn't make the link (and of course neither does West, since he doesn't make that argument). In Athenaeus we have a passage where he categorises various types of tragic actors, and one of the categories he mentions is called a magōidos, who specialises in playing both (male and?) female roles (620e; but the text seems to be corrupt), and who "holds drums and cymbals and is entirely dressed in women's clothing" (621c). He attributes the first bit to the 4th-century-BCE musical theorist Aristoxenus. The word magōidos would literally mean "magos-singer". I'm sure if Rozik were aware of this, he'd use it as part of his argument. But though there are a few isolated parallels between the magoi and the Orphic version of Dionysus, on the one hand, and Asian "shamans" on the other, there's certainly no direct link, and no real support for an indirect link.