I bought a product called Kayoko and it runs on Zend Optimizer, my infrastructure in on Windows Azure Web Sites! Azure web sites says it runs on PHP 5.3 and 5.4! can it run on Windows Azure Web Sites?
If what you are talking about is this. It clearly shows that it requires PHP 5.3. So you can run it, respecting the PHP version.
Now, the Zend optimizer is a bit tricky to run, but not impossible. Check out this blog post on how to configure third party PHP extensions. And this StackOverflow question and my answer on how to run IonCube which is similar to ZendOptimizer.
And please update your question to include concrete product names and appropriate links.