How to email user after inserting the username in database in WordPress

Assuming that $member_details->user_login is already a user in the wp_users table, then you could use the following to get their email address:

$user = get_user_by( $member_details->user_login, 'login' );

From there, you have the user’s email address and can use wp_mail() to email them:

$subject = "My email subject";
$message = "My message body...";
wp_mail( $user->user_email, $subject, $message );