No, there is no way to submit answers without logging in. The "anonymous" feature is just a setting to hide your username from the leaderboards.

You have to have a login (Google, Github, Reddit, Twitter) to get inputs, and part 2 is only unlocked after completing (correctly) part 1 for any given day. You don't have to share your identity with anyone but the host of the server.

You have to log in, but you don’t have to reveal your identity or username on the leaderboard.

Other comments haven’t pointed this out, but the key thing is that each account gets a different input (so it's a little harder to cheat).

It's easy to cheat by applying someone else's code (ex: from reddit) to your own input.