I think the easiest way to change one of Social Network to VKONTAKTE, for example in phrase system search for ICQ or AOL IM and change the phrase to Vkontakte. This will be displayed as Vkontakte field on forum front.

For example you change AOL IM to Vkantacte.

$field = 'aim';

For example you change ICQ to Vkantacte.

$field = 'icq';

You should add a small code in wpForo Field hook. Navigate to Dashboard > Appearance > Editor and put this code in functions.php file of your active child theme's functions.php file. If it doesn't have child theme put it in current theme functions.php file.

I'd also recommend use WP Editor plugin instead of WordPress native editor. The native editor sometimes doesn't save changes. Just install WP Editor then navigate to the same page and put this code as described above.

function wpforo_change_social_field_icon( $fields ){
 //Field name
 $field = 'aim';
 //Font-Awesome icon class  http://fontawesome.io/icons/ 
 $icon = 'fa-vk';
 if( !empty($fields) && isset($fields[$field])) $fields[$field]['faIcon'] = $icon;
 return $fields;
}
add_filter( 'wpforo_member_after_init_fields', 'wpforo_change_social_field_icon' );

The functions.php file is not the same for all themes, the content code is different, but all extra codes people insert in functions.php file. Just click on line #4, copy the code and paste there. Then save it and check. If you doubt, just send admin login details to support[at]gvectors.com email address, we'll do that for you.

Do we do this for every social network we need to change?

For example, I am changing AIM for Instagram, ICQ for Flickr and MSN for YouTube.

I have done it for Instagram already, but when filling out the field, it doesn't ask for a URL so the result isn't a link (like the Twitter field).

And how do we remove the other fields we don't need?