My first step would be to remove the dependency on jQuery. I'm not sure why a whole library would be needed if all you're doing is a web form.
This should totally work. Trying again is the second suggested step. An HTML widget is basically an iFrame...
...but an iFrame can be added manually with HTML code in an element.
A browser debugger / Safari Web Inspector can be used to find the error.
Theoretically, the form could be created in HTML in Hype. Your form processor on the server would just have to be able to handle it. I say theoretically, as bad PHP code here can be a career killer. If you don't sanitize the inputs properly, your server could be compromised. That's why I don't use Hype for this. I just send people directly to the form.