Add Share option without an app? Is it possible to install/setup a new sharing option without the use of an app?

In my case I'd like to be able to share photos to Flickr, but I have no interest in installing the Flickr app (or any other app whatsoever, for that matter).
No. It's not just a question of adding something to the chooser (the menu of apps): there needs to be an app on the other end to take the photo, log into Flickr, and understand Flickr's API (the programming interface that apps use) in order to upload the photo. It also needs to handle possible failures, such as resuming or retrying the upload later if you lose internet connection.

If you want a share option that shares to Flickr, there are many Flickr apps on Google Play that only do uploading, and don't have any functionality to browse or view photos. That way you can have the share option while still keeping your phone lean.

Alternatively, you can browse to the Flickr website in your web browser and upload photos from there. It's not as convenient but it's the only way that doesn't involve any kind of special-purpose app.