Using wp.editor in widgets

README.md

Within your widget, you can include a editor with a textarea and the class custom-widget-wp-editor.

 <textarea id="<?php echo $this->get_field_id( 'header' ); ?>" name="<?php echo $this->get_field_name( 'header' ); ?>" class="custom-widget-wp-editor"><?php echo $header; ?></textarea>

Make sure you include the JS-file within the widgets and customizer view.

/**
 * Add a fix for tinymce within widgets.
 *
 * @return void
 */
function custom_widgets_widget_editor_script() {
  global $pagenow;

  if ( 'widgets.php' === $pagenow || 'customize.php' === $pagenow ) {
    wp_enqueue_editor();

    wp_enqueue_script( 'wp-editor-widgets', get_stylesheet_directory_uri() . '/js/wp-editor-widgets.js', array( 'jquery' ), '1.0', false );
  }
}

add_action( 'admin_enqueue_scripts', 'custom_widgets_widget_editor_script' );

wp-editor-widgets.js

jQuery(document).ready(function ($) {

    /**
     * Removes all active editors. This is needed so they can be
     * re-initialized after a widget-updated event.
     */
    function custom_widgets_remove_editors() {
        custom_widgets_get_editors().forEach(function (id) {
            wp.editor.remove(id);
        });
    }

    /**
     * Searches the widgets areas for editors.
     *
     * @return array
     */
    function custom_widgets_get_editors() {
        let editors = [];

        // Doesn't look so great, but we need to target like this to support both the regular
        // widgets view and the Customizer. Also, we don't want to include the placeholder
        // widget that is in the DOM on the widgets.php page.
        $(document).find("#customize-theme-controls .custom-widget-wp-editor, #widgets-right .custom-widget-wp-editor").each(function () {
            editors.push($(this).attr('id'));
        });

        return editors;
    }

    /**
     *  Initializes all wp.editor instances.
     */
    function custom_widgets_set_editors() {
        custom_widgets_get_editors().forEach(function (id) {
            wp.editor.initialize($('#' + id).attr('id'), {
                tinymce: {
                    wpautop: true
                },
                quicktags: true,
                mediaButtons: true
            });
        });

        // To prevent the editor from not submitting the value, we click the switch html tab.
        $(document).contents().find('.widget-control-save').off().on('click', function (e) {
            custom_widgets_get_editors().forEach(function (editor) {
                let form = $('#' + editor).closest('form');
                form.find('.switch-html').click();
            });
        });
    }

    // Trigger removal and setting of editors again after update or added.
    $(document).on('widget-updated widget-added', function () {
        custom_widgets_remove_editors();
        custom_widgets_set_editors();
    });

    // Init.
    custom_widgets_set_editors();
});

Thanks for sharing @molcsa!