Is it possible to do something so it doesn't try to grab image and simply sets default avatar?
Yep you can set a default avatar from wp-admin -> wplms -> Buddypress -> " Upload BuddyPress default member avatar "and save changes
refer :
http://prntscr.com/d7wqht
I had the avatar enabled but when users register or login with social media avatar is not pulled. Just broken image....
The broken image indicates that the PHP in your server is not able to grab the image from the link.
This happens when allow_url_fopen = On is not set in php.ini of your server.
I am adding a check for this in your site,so that this issue can be resolved.
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Please wait for ym update
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I have added a fix please check it again with a new facebook user account .