Sure, but they don't have the "home field advantage" that the NSA does, whereby much of the core internet infrastructure is housed in the US. I forget the exact number, but something like 70% of the world's internet traffic transits the US. (they mention this constantly in NSA-related articles)

But how much of the core infrastructure is made by the Chinese?

An astute question, considering the strange wifi-chips-in-irons story.

Are they? That's a very complicated thing to pull off and China isn't known for having the most advanced Navy (e.g. they can't secure their own oil tankers in the Persian Gulf) and they only have a small number of submarines.