This php script shows how you search for a customer before creating the case, however it doesn't plan for the customer not existing, you'd have to check the response for total_entries == 0 or an empty entries array and create the customer accordingly.

search_customer_create_case.php

<?php

$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, "https://yoursite.desk.com/api/v2/customers/search?email=thomas%40stachl.me");
curl_setopt($ch, CURLOPT_PORT, 443);
curl_setopt($ch, CURLOPT_TIMEOUT, 15);
curl_setopt($ch, CURLOPT_USERAGENT, $_SERVER['HTTP_USER_AGENT']);
curl_setopt($ch, CURLOPT_USERPWD, "email:password");
curl_setopt($ch, CURLOPT_HTTPAUTH, CURLAUTH_BASIC);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "GET");
 
$customer_search_result = json_decode(curl_exec($ch));
$customer_path = $customer_search_result->_embedded->entries[0]->_links->self;

$case = array(
  'type' => 'email',
  'subject' => 'Email Case Subject',
  'priority' => 4,
  'status' => 'open',
  'labels' => array('Spam', 'Ignore'),
  'message' => array(
    'direction' => 'in',
    'body' => 'Example Body',
    'to' => 'support@example.com',
    'from' => 'jdoe@example.com',
    'subject' => 'My email subject'
  ),
  '_links' => array(
    'customer' => $customer_path
  )
);

$case_string = json_encode($case);

echo $case_string;
echo "\n\n";
 
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, "https://yoursite.desk.com/api/v2/cases");
curl_setopt($ch, CURLOPT_PORT, 443);
curl_setopt($ch, CURLOPT_TIMEOUT, 15);
curl_setopt($ch, CURLOPT_USERAGENT, $_SERVER['HTTP_USER_AGENT']);
curl_setopt($ch, CURLOPT_USERPWD, "email:password");
curl_setopt($ch, CURLOPT_HTTPAUTH, CURLAUTH_BASIC);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "POST");
curl_setopt($ch, CURLOPT_POSTFIELDS, $case_string);
curl_setopt($ch, CURLOPT_HTTPHEADER, array(
    'Content-Type: application/json',
    'Content-Length: ' . strlen($case_string)
  )
);
 
$return = curl_exec($ch);
echo $return;
exit;