I am unaware of any part of WebAPI which is geared toward native code performance. And cool as it is, asm.js is never going to have the same potential that PNaCl has. Primarily this falls under one reason: multi-threading.

Web Workers are a step in the right direction, but they're incredibly limited by design because of the inability to share memory across threads. The requirement of copying data across threads means that parallelizable tasks involving small operations over large chunks of data often don't benefit from using Web Workers. Unfortunately those tasks make up a large percentage of situations where parallel processing is useful.