search customers in front end by billing data

I think you have two issues that can be resolved in one go:

'search' => '*' . esc_attr( $search_term ) . '*',

• You don’t need the '*'s around the term. It looks like you’re trying to do a wildcard, but that’s what search already does by adding SQL ‘%’ around your term. Essentially, you end up searching for "%*{$search_term}*%", which is probably not what you want.
• esc_attr should be unnecessary here. The WordPress database class used in your query will escape your input for you.
• Using search in this context is creating a query like this:

SELECT ... FROM wp_users JOIN wp_usermeta ... WHERE (
 (
   meta_key = 'nickname' AND meta_value LIKE 'search term'
 ) OR 
 (
   meta_key LIKE 'last_name'
 ...
 )
AND
 (
   wp_users.ID LIKE 'search term'
  OR
   wp_users.user_login LIKE 'search term'
 )
) -- END WHERE

which is also probably not what you want. In a WP_User_Query, search means “Search for user logins matching this query”, not “search for general user meta matching this query.”

So, try something like this:

    // wp_user_query arguments
    $args = array (
        'role'       => 'customer',
        'order'      => 'asc',
        'orderby'    => 'display_name',
        'meta_query' => array(
            'relation' => 'or',
            array(
                'key'     => 'nickname',
                'value'   => $search_term,
                'compare' => 'like'
            ),
            array(
                'key'     => 'last_name',
                'value'   => $search_term,
                'compare' => 'like'
            ),
            array(
                'key'     => 'billing_phone',
                'value'   => $search_term,
                'compare' => 'like'
            )
        )
    );