As an administrator, I can see and edit the user’s avatar. As a restricted administrator (i.e. assigned a custom role), I am unable to see it.
Here’s a link to the plugin’s screenshot showing you the permissions page: https://s.w.org/plugins/members/screenshot-3.png?r=879755
Basically, the avatar section is not listed for me as an option so I’m assuming I would need to add it as a custom capability using the field that is displayed at the bottom of the screen shot.
Does that help?