you can use ajax from Phtml file to the controller to get JSON Data. you controller should return JSON response.

Code for controller returning JSON response given below.

namespace Company\MyModule\Controller\Ajax;

\Magento\Framework\Controller\Result\JsonFactory $resultJsonFactory,  

class Index extends \Magento\Framework\App\Action\Action
{
 
.......
private $resultJsonFactory;
.......

public function __construct(
........
\Magento\Framework\Controller\Result\JsonFactory $resultJsonFactory,
........
){
 
......
$this->resultJsonFactory   = $resultJsonFactory;
......
}
 public function execute()
{
 $result = $this->resultJsonFactory->create();
$jsonResponse = [
                'Field_1'  => "value 1",
                'Field_2' => "value 2"
            ];
return $result->setData($jsonResponse);
}

You can use session on your observer to store JSON response that you are getting from API.

Later on the phtml, you get this session data from a Block file.

you can use below code in your observer to store data in the customer session

observer code

 <?php
namespace Vendor\Module\Observer\Customer;    

class Authenticated implements ObserverInterface {
    ............
    protected $_customerSession;
    ............

    public function __construct(
            ............
            \Magento\Customer\Model\Session $customerSession
            ............
    ) {
        ............
         $this->_customerSession = $customerSession;
         ............
    } 

    public function execute(\Magento\Framework\Event\Observer $observer) 
    {
        ............
        $this->_customerSession->setMyVar1('my var 1 data');
        $this->_customerSession->setMyVar2('my var 2 data');
        ............
        return $this;
    }
}

Block class code (assumed your Block class file is Vendor\Module\Block\Myblock.php)

<?php
 
namespace Vendor\Module\Block;

class Myblock extends \Magento\Framework\View\Element\Template
{
    
    private $customerSession;
    private $storeManager;    
    
    public function __construct(
        \Magento\Framework\View\Element\Template\Context $context,    
        \Magento\Customer\Model\Session $customerSession,      
        \Magento\Store\Model\StoreManagerInterface $storeManager
         
    ) {        
        $this->customerSession = $customerSession;        
        $this->_storeManager = $storeManager;                
        parent::__construct($context);
    }
    
    
    public function getMyVarOne() 
    {
        return $this->getCustomerSession()->getMyVar1();
    }
    
    public function getCustomerSession() 
    {
       return $this->customerSession;
    }    
}

Phtml File Code (assumed your phtml file is Vendor/Module/view/frontend/templates/myblock.phtml)

<?php echo $block->getMyVarOne(); ?>

Please Note: since you will store JSON data, you may need to use json_encode and json_decode to store and parse data.