I suggest revising the eveRequest class as follows:

class eveRequest {
  // User Data
  protected $userID = NULL;
  protected $apiKey = NULL;
  protected $isFullKey = NULL;     // Added
  // Api method
  protected $requestURI = NULL;
  protected $fullKeyRequired = NULL;    // Added
  private   $baseURI = 'http://api.eve-online.com/';

  // Post request data
  protected $data = NULL;            // Changed to protected

  private $cachedUntil = NULL;
  private $currentTime = NULL;

  public function __construct($userID, $apiKey, $reqURI, $fullRequired) {
    $this->userID = $userID;
    $this->apiKey = $apiKey;
    $this->$requestURI = ( isNull($reqURI) ? NULL : $reqURI );          // Added
    $this->$fullKeyRequired = ( isNull($fullRequired) ? false : $fullRequired );     // Added
  }

$isFullKey and $fullKeyRequired are added to accomodate requests that require a full API key rather than a limited one.
$data is changed to protected as derived child classes will need ready access to it
__construct() had 2 additional arguments added permitting the type of request to be established on object creation.

These changes seem advisable because of the wealth of child classes that may be derived from this simple class. See this page for examples.