This turns out to be one of the Great Secrets of the Internet, and I wasted around half an hour scanning documentation, Googling various permutations of "NuGet API Key" and "How to find NuGet API Key" before finally finding Phil Haack's page here: http://haacked.com/archive/2011/01/12/uploading-packages-to-the-nuget-gallery.aspx/
Anyway, if you're in the same boat as me and desperately trying to figure out how to get your API key for a local NuGet feed:
- Sign in to the web front end on your local NuGet server
- Click your username (somewhere on the top right, usually)
- The API key is displayed somewhere near the bottom in the "Credentials" section.
Another (small) mystery solved!