<h:inputText>doesn't have "onsubmit" event. Use a4j:form.
it doesn't work for me with a4j:form because I don't have a button or link too submit.
And it seems to work fine ...
You can use a4j:form to generate AJAX requests for common form submits. If that's not what you want, you'll have to define "onsubmit" a4j:support as direct form child.