Their documentation is not clear at all
There is nothing at all anywhere on the site labeled as the "API Key" but there is
You can get the consumer_key and consumer_secret after registering a new app at Tumblr here (it is called OAuth Consumer Key and there is a link below to get the Secret key of the app).
The token and token_secret keys are generated after allowing access of an app to a Tumblr account. They will be returned as parameters after returning to the callback script.