I've been asked to create a simple web part that will create a post in a specific yammer group.
All the code samples I have found involve registering an app. Is an app the only way?
In order to interact with the Yammer API, you need to create an app in Yammer. It's a very simple process and the purpose is to secure an ID and secret so that you can authenticate against the Yammer API. It's not an app in the SharePoint sense (or O365). Check out the documentation here https://developer.yammer.com/v1.0/docs/app-registration Once this is done, then you can use JS SDK to work on your project.
Also, depending on what you need the web part to do, you may just be able to use Yammer Embed (which can be put into a script editor web part or content query web part), and allows you to post to a group, as well as display the feed from the group. https://developer.yammer.com/v1.0/docs/embed