It seems like currently the only way to display avatars is via Gravatar:

This is less than ideal since:
1) Gravatar now requires a WordPress sign-up. The sign-up process is a bit buggy as I had some issue with Gravatar's linking to Gravatar's website, ultimately leading to me having to contact their support (no response yet).
2) Though a lot of users have gravatar from many years back, a sign-up is a lot friction for new users to have to deal with.
3) Sometimes people may want to use a different avatar than their e-mail's main avatar.
4) Some users may have privacy concerns of gravatar now being able to harvest all the different users of various websites.

Please add support for local avatars