<form enctype="multipart/form-data"  id="form" action="send.php"  method="post">
      <div class="modal-body d-flex flex-column">
        <input class="mb-4" type="text" id="name" name="name" placeholder="Имя" required>
        <input class="mb-4" type="text" id="namber" name="number" placeholder="Телефон" required>
      </div>
      <div class="modal-footer justify-content-center">
        <button form="form" class="mb-4 main-btn send-btn">Записаться</button>
      </div>
    </form>

$('#form').on('submit', function(e){
   e.preventDefault();
   var fd = new FormData(this);
   $.ajax({
    url: 'send.php',
    type: 'POST',
    contentType: false,
    processData: false,
    data: fd,
    success: function(msg){
     if(msg == 'ok') {
     form.reset();
     $(".close").click();
     } else {
     alert("Ошибка");
     }
    }
   });
  });

<?php

use PHPMailerPHPMailerPHPMailer;
require 'PHPMailer.php';
require 'SMTP.php';

$name = $_POST['name'];
$number = $_POST['number'];

$mail = new PHPMailer;

$mail->isSMTP();
$mail->Host = 'smtp.timeweb.ru';
$mail->SMTPAuth = true;
$mail->Username = 'zayavka@*****.ru';
$mail->Password = '*****';
$mail->SMTPSecure = 'ssl';
$mail->Port = 465;
$mail->setFrom('zayavka@*****.ru');
$mail->addAddress('****@yandex.ru');

$mail->isHTML(true);
$mail->Subject = "Заявка";
$mail->Body = "Имя $name . Телефон $number";

if($mail->send()) {
    echo 'ok';
} else {
    echo 'Message could not be sent.';
    echo 'Mailer Error: ' . $mail->ErrorInfo;
}

?>