Blesta doesn't currently have a generic domain availability checker for you to integrate with in other systems. Currently, domain lookups are done using a specific module in Blesta (e.g. Enom).
What you can do is use the Blesta API SDK to call ModuleManager::moduleRpc to check a domain's availability, e.g.:
// Load the API SDK
require_once "blesta_api.php";
$user = "username";
$key = "key";
$url = "https://yourdomain.com/installpath/api/";
$api = new BlestaApi($url, $user, $key);
// Choose one of the domain registrar modules from Blesta to use to lookup domain availability
$module_id = 1; // set to the module ID of the domain module you want to use, like Enom
$module_row_id = 1; // set the specific module row ID from the module that you want to use
$domain = "google.com"; // the domain to check for availability
$response = $api->get(
"ModuleManager",
"moduleRpc",
['module_id' => $module_id, 'method' => 'checkAvailability', 'params' => [$domain], 'module_row_id' => $module_row_id]
);
// View the response from the check
if ($response->response() === true) {
// The domain is available
} else {
// The domain is not available
}