UPDATE: the latest Azure release (specifically, the Mobile Services backend) supports arbitrary NPM modules - See the section on Mobile Services: NPM Module Support
From the JSDom docs on the npm:
Unfortunately, doing this kind of magic requires C++. And in Node.js, using C++ from JavaScript means using "native modules." Native modules are compiled at installation time so that they work precisely for your machine; that is, you don't download a contextify binary from npm, but instead build one locally after downloading the source from npm.
https://npmjs.org/package/jsdom#contextify
Since your "native modules" don't have the same signature as whatever machine in azure's "native module" would be signed, it doesn't load.