Magento 2 – Customer Session for Logged In Customer with Token-Based Authentication

What I understand from you question is, You have only email (and obviously some network specific secret key to validate i.e facebook key).

So You just need to load the customer by email id as below.

protected function getCustomerToken($emailId){
 /**
* @var \Magento\Customer\Model\Customer $customer */
*/
$customer->loadByEmail($emailId);
if($customer->getId()){
        /**
        * @var \Magento\Integration\Model\Oauth\TokenFactory $tokenModelFactory 
        */
        $customerToken = $this->tokenModelFactory->create();
        $tokenKey = $customerToken->createCustomerToken($customerId)->getToken();
        return $tokenKey;
}
return "YOU MSG FOR CUSTOMER NOT FOUND";
}

The above code should return the token key without password.

Note: Make sure you are doing proper & strong validating before generating the token & rest is already explained in Franck's answer .

Yes. you can use Magento 2 API this way but you need to create a php page (say php page name is apicall.php) and make ajax call to that custom page instead Magento directly. On the the custom.php page you need to call Magento API.

apicall.Php

<?php
// REPLACE WITH YOUR ACTUAL DATA OBTAINED WHILE CREATING NEW INTEGRATION
define("CONSUMERKEY", "XXXXXXXXXXXXXXX");
define("CONSUMERSECRET", "XXXXXXXXXXXXXXX");
define("ACCESSTOKEN", "XXXXXXXXXXXXXXX");
define("ACCESSTOKENSECRET", "XXXXXXXXXXXXXXX");         


function sign($method, $url, $data, $consumerSecret, $tokenSecret)
{
    $url = urlEncodeAsZend($url);

    $data = urlEncodeAsZend(http_build_query($data, '', '&'));
    $data = implode('&', [$method, $url, $data]);

    $secret = implode('&', [$consumerSecret, $tokenSecret]);

    return base64_encode(hash_hmac('sha1', $data, $secret, true));
}

function urlEncodeAsZend($value)
{
    $encoded = rawurlencode($value);
    $encoded = str_replace('%7E', '~', $encoded);
    return $encoded;
}


$actionName = $_REQUEST['action']; 

switch($actionName){
     case 'getcustomer':
      echo getCustomById();
      break;
}      

function getCustomById()
{

$customerId =  $_REQUEST['custid'];
$result =  null;
$method = 'GET';
$url = 'YOUR-MAGENTO-ROOT/index.php/rest/V1/customers/'.$customerId;

//
$data = [
    'oauth_consumer_key' => CONSUMERKEY,
    'oauth_nonce' => md5(uniqid(rand(), true)),
    'oauth_signature_method' => 'HMAC-SHA1',
    'oauth_timestamp' => time(),
    'oauth_token' => ACCESSTOKEN,
    'oauth_version' => '1.0',
];

$data['oauth_signature'] = sign($method, $url, $data, CONSUMERSECRET, ACCESSTOKENSECRET); 
$curl = curl_init(); 
curl_setopt_array($curl, [
    CURLOPT_RETURNTRANSFER => 1,
    CURLOPT_URL => $url,
    CURLOPT_HTTPHEADER => [
        'Authorization: OAuth ' . http_build_query($data, '', ',')
    ]
]); 
$result = curl_exec($curl);

     return json_encode(array(             
            'result' => $result
        ));
}

calling Ajax to get custom data:

<script type="text/javascript">
        $(document).ready(function(){       
            $.getJSON('YOUR-MAGENTO-ROOT/apicall.php?action=getcustomer&custid=1', function(jd) {
                 var result =jQuery.parseJSON(jd.result);
                 console.log(result);
                 jQuery.each(result, function(index, value) {
                     console.log(index+":"+value);
                 });

               });
        });
    </script>