On my Khan Academy app on iPad, I cannot change it either; however, you can still do so by signing into your account on khanacademy.org on your chrome/safari. It will give you the same webpage as if it were on your laptop. 

Hi Cody,

I'm afraid there's not currently a way to change your avatar through the mobile app---this can only be done through the desktop version of the site. You can find instructions for changing your avatar here: https://support.khanacademy.org/hc/en-us/articles/115007749488-How-do-I-change-my-avatar-

However, as Christina said, you can still accomplish this using the internet browser (Chrome, Safari, etc.) on your mobile device.

I hope this helps! Please let me know if there's anything else I can do for you!