The avatar upload is in the profile tab of the settings page which is different than your profile page just to keep you on your toes.

@SustainedChaos You go to profile dropdown menu-> Settings -> profile -> (scrolldown to) avatar. Select a file and press Save.

‘wait, Reddit has avatars?’ - said swathes of users who discovered Reddit added useless features to their site making reddit.com look like absolute shit compared to old.reddit