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You can replace this code

1. Controller Search.php

<?php

namespace Vendor\SearchStores\Controller\Index;

use Magento\Framework\App\Action\Action;
use Magento\Framework\App\Action\Context;
use Magento\Framework\View\Result\PageFactory;
use Magento\Framework\Registry;
use Magento\Framework\Controller\Result\JsonFactory;

//use Magento\Framework\Controller\Result\Forward\Factory;

class ReadSearch extends Action
{
    protected $resultPageFactory;

    protected $resultJsonFactory;

    protected $_coreRegistry;

    public function __construct(
        Context $context,
        PageFactory $pageFactory,
        Registry $coreRegistry,
        JsonFactory $resultJsonFactory
    ) {
        $this->resultPageFactory = $pageFactory;
        $this->_coreRegistry = $coreRegistry;
        $this->resultJsonFactory = $resultJsonFactory;
        parent::__construct($context);
    }

    public function execute()
    {
        $searchWord = $this->getRequest()->getParam('search');
        if ($this->_coreRegistry->registry('search_word')) {
            $this->_coreRegistry->unregister('search_word');
        }
        // set value to _coreRegistry variable  search_word
        $this->_coreRegistry->register('search_word', $searchWord);
        $resultPage = $this->resultPageFactory->create();
        $result = $this->resultJsonFactory->create();
        $block = $resultPage->getLayout()
            ->createBlock('Vendor\SearchStores\Block\ShowResult')
            ->setTemplate('Vendor_SearchStores::results_modal.phtml')
            ->toHtml();
        $result->setData(['output' => $block]);

        return $result;
    }
}

2. Block ShowResult.php

<?php

namespace Vendor\SearchStores\Block;

use Magento\Framework\View\Element\Template\Context;
use Magento\Framework\View\Element\Template;
use Magento\Framework\Registry;

class ShowResult extends Template
{
    protected $_coreRegistry;

    public function __construct(
        Context $context,
        Registry $coreRegistry
    ) {
        $this->_coreRegistry = $coreRegistry;
        parent::__construct($context);
    }

    public function getSearchWord()
    {
        $searchWord = '';
        if ($this->registry->registry('search_word')) {
            $searchWord = $this->registry->registry('search_word');
        }
        //HERE IS WHERE I WANT TO GET THE DATA FROM THE CONTROLLER
        return $searchWord;
    }
}

After making the above changes in your Controller and Block file, you can get a search word in your template file.

Hope this will help you!