Is it possible to shorten a URL using the Goo.gl shortening api with a GET request? Their only instructions are for POST and it doesn't make much sense that they wouldn't have a way to do this via GET.
It's actually unlikely that they support GET to do that. Good practice requires that GET requests not cause side effects (permanent data changes) in web applications. This prevents problems related to web spiders causing havoc simply by trying to crawl a site (imagine a "delete" button that worked with a GET, causing a spider to inadvertently remove content).
Additionally, GET requests are a lot easier to force a third party to do (i.e. embed the url in an image tag on a forum) which often is a security problem. In the case of goo.gl, it would allow trivial and hard to block DoS type attacks on the service.