Since Customizr v3.5+

Since Customizr v3.5, you can add a social link by adding this type of code to your functions.php :

Before Customizr v3.5+

1) Add them!

Note : Tumblr now already included in the default Customizr social links. In your child theme functions.php add this simple snippet:

add_filter('tc_default_socials', 'add_new_socials', 20);
function add_new_socials($socials){
    $new_socials = array(
        //tc_yoursocial => array
        'tc_tumblr' => array(
            'link_title'    => __( 'Follow me on Tumblr'), //title displayed on mouse hover
            'option_label'  => __('Tumblr profile url'),
            'default'       => "http://www.tumblr.com" //your social link here
        ),
        'tc_delicious' => array(
            'link_title'    => __( 'Find me on Delicious'),
            'option_label'  => __( 'Delicious profile url' ),
            'default'       => "http://delicious.com/"
        ),
    );

    return array_merge( $socials, $new_socials );
}

Read How to customize the Customizr WordPress theme? if have never used a functions.php file in a child theme.

 2) – Let’s style them!

Note : Since Customizr 3.4.19 and Customizr-Pro 1.2.19 the theme moved to the Font Awesome icon set so to refer to its map to get the correct code to use in the ‘content’ attribute.
Add this in your child theme style.css. Note : In the code below, I use one backslash for the content property value. If you want to use this css in the Custom CSS in customizer you have to use two backslashes because you need to “escape” one. Since Customizr 3.3.1 (Customizr-Pro 1.0.12), because of new wp themes guidelines,  you cannot use single (double) quotes in the Custom CSS in customizer. You need a child-theme.
Note : When you use a custom image as icon you might need to resize it for different viewports. To achieve that you have to use  media queries.

/* Thumblr */
.icon-tumblr:before {
    content: "\f214"; /* we can use a genericon for tumblr */
}

/* Delicious */ 
.icon-delicious:before {
    content: "";
    width: 18px;
    height: 18px;
    display: inline-block;
    background: url("https://upload.wikimedia.org/wikipedia/commons/1/11/Delicious.svg") no-repeat;
    background-size: 18px 18px
}

/* change delicious icon size in sidebars*/
.social-block.widget_social .icon-delicious:before {
    width: 14px;
    height: 14px;
    background-size: 14px 14px;
}

/* change delicious icon size in footer */
#footer.footer .colophon .icon-delicious:before {
    width: 16px;
    height: 16px;
    background-size: 16px 16px;
}

Enjoy!