A very simple, yet complete, AJAX contact form for WordPress.

ContactFormProcessing.php

<?php 
class ContactFormProcessing
{
    protected $action = 'example_contact_form';

    public function ajaxInit()
    {
        add_action("wp_ajax_{$this->action}", [$this, 'process']);
        add_action("wp_ajax_nopriv_{$this->action}", [$this, 'process']);
    }

    /**
     * Declare the form fields
     *
     * @return array
     */
    public function fields()
    {
        return [
            'name'    => [
                'label' => 'Name',
                'type'  => 'text',
            ],
            'email'   => [
                'label' => 'Email address',
                'type'  => 'email',
            ],
            'message' => [
                'label' => 'Message',
                'type'  => 'textarea',
            ]
        ];
    }

    public function render()
    {
        $output = '<form class="' . $this->action . '" method="post">';

        foreach ($this->fields() as $fieldId => $field) {
            $value = !empty($field['default']) ? $field['default'] : '';

            $output .= "<div class='form-field form-field--{$fieldId}'>";
            $output .= "<label>{$field['label']}</label>";
            switch ($field['type']) {
                case "email":
                    $output .= "<input type='email' value='" . esc_attr($value) . "' name='{$fieldId}' />";
                    break;
                case "textarea":
                    $output .= "<textarea name='{$fieldId}' />" . esc_textarea($value) . "</textarea>";
                    break;
                default:
                    $output .= "<input type='text' value='" . esc_attr($value) . "' name='{$fieldId}' />";
                    break;
            }
            $output .= "</div>";
        }

        $output .= "<button type='submit'>Send message</button>";
        $output .= "</form>";

        return $output;
    }

    /**
     * Process the AJAX request.
     *
     * @return string The JSON string.
     */
    public function process()
    {
        // prepare a generic response.
        $response = [
            '_status'  => 'success',
            '_message' => 'Your message has been sent. Thank you.'
        ];

        // Get the data from AJAX
        if (!empty($_POST['data'])) {
            // The data is sent as a URL string. Convert it to array
            parse_str($_POST['data'], $data);

            // Just loop over the form fields
            foreach ($this->fields() as $fieldId => $field) {
                // Process this individual field.
                if (empty($data[$fieldId])) {
                    $response['_status']  = 'error';
                    $response['_message'] = "The `{$field['label']}` is required.";
                    break;// Stop the loop. We have an error.
                } else {
                    // Save data somehow if needed.
                }
            }

            // Send the email if all input data is OK.
            if ($response['status'] == 'success') {
                wp_mail(
                    'my_mail@example.com',
                    'You got a new message.',
                    $data['message'],
                    'From: ' . $data['email'] . "\r\n" . 'Reply-To: ' . $data['email'] . "\r\n"
                );
            }
        }
        // Return the response as a JSON string that can be parsed in JavaScript.
        echo json_encode($response);

        die();
    }
}

scripts.js

jQuery('.example_contact_form').on('submit', function (event) {
    event.preventDefault();

    var data = jQuery(this).serialize();

    jQuery.ajax({
        type: "POST",
        url:  "wp-admin/admin-ajax.php",

        data: {
            'action': 'example_contact_form',
            'data':   data
        },

        success: function (response) {
            // console.log(response);
            data = JSON.parse(response);

            if(data._status === 'success'){
                jQuery(':input').val(''); // Clear the form
            }

            alert(data._message);
        },

        complete: function () {
        },
    });
});

Init the contact form in functions.php:

$cf = new ContactFormProcessing();
$cf->ajaxInit();

Place this code in that place where you want to display the form(ie: to the end of a blog post).

$cf = new ContactFormProcessing();
echo $cf->render();