The Tumblr API supports basic content reading, but I have not yet found a way to get either the list of followers, or the list of people you are following. The API seems to only return the follower count. The theme documentation has a block that can show who you follow, which is the closest I have found to what I want.
My question is: Given a user's email address and password, is it possible to find either of these lists, through any means? Some sort of background job might be necessary.
To answer my own question:
While this was not possible with the Tumblr API before, the newly release API v2 now supports both types of information.
/followers and /user/following are the relevant commands. They return JSON objects which list the blogs that follow a user, or that a user follows. It looks like they also require OAUTH to access, as this is not public information.