Hi Rachel,
I have a self-use only translation extension, that supports many translate engines including MS Translator API. That api certainly can work in chrome extension, no problem, because it is a general REST service, and you added host url in "permissions" field of manifest.json.
Some suggestions,
First, you should do a basic test, in any extension page(not content script) fetch that api.
405000: The request method is not supported for the requested resource.
Maybe, you should check your Azure service dashboard, the resource is enabled?