Currently, form fields do not work in a Publish Online document. To collect data from a form submission, you would need to direct the user to an external source. Here is a solution that you can easily create, publish, and collect all submissions from a form.
Ok thanks Jeffrey. Hopefully the ability to embed simple surveys like a pulse survey right on an Adobe Publish Online page and be able to program it to collect answers somewhere easy would be cool. I understand that's probably a more complicated undertaking that it sounds.