Add a URL field to the WordPress Customizer.

WP Customizer - URL

$wp_customize->add_setting( 'themeslug_url_setting_id', array(
  'capability' => 'edit_theme_options',
  'sanitize_callback' => 'themeslug_sanitize_url',
) );

$wp_customize->add_control( 'themeslug_url_setting_id', array(
  'type' => 'url',
  'section' => 'custom_section', // Add a default or your own section
  'label' => __( 'Custom URL' ),
  'description' => __( 'This is a custom url input.' ),
  'input_attrs' => array(
    'placeholder' => __( 'http://www.google.com' ),
  ),
) );

function themeslug_sanitize_url( $url ) {
  return esc_url_raw( $url );
}