When someone is logged into this member area, are they also logged into WordPress?
The ajax call that retrieves the icons for the select2 field assumes a logged in WordPress user (the wp_ajax_ call). There could potentially be another call added for unauthenticated users (the wp_ajax_nopriv_ call). I would have to think about any potential side effects/ramifications of the change.
A quick test would be: does this field work for you if you are logged in to your WordPress admin account?
The potential fix to this would be adding this line:
add_action( 'wp_ajax_nopriv_acf/fields/font-awesome/query', array( $this, 'select2_ajax_request' ) );
right underneath line #65 in /assets/inc/class-ACFFA-Loader-5.php. Or under line #42 in /assets/inc/class-ACFFA-Loader-4.php if you are on FontAwesome 4.