Add a new section under Settings for your social accounts. The order and the images can be customized with ease.
In the settings at the top of the page, change the Visible accounts to All Accounts
When you hover the title of the accounts, an move pointer will show, you can then simply drag and drop.
Click on the icon, the WordPress media manager popup will show and you'll be able to select the new icon to use.
An x will be visible at the top right corner of the custom image. Click on it to revert back to the default image.
<?php if (function_exists('the_social_accounts')) the_social_accounts(); ?> in the the template files.