You're right the submit with a module is a bit more difficult because form submits have to go to a URL. It is messier to have to write a component just to accept the input from the module too.
Because of this issue, in Joomla 3 is Matt Thomas wrote a component called com_ajax. Com_Ajax is a component that is designed to assist modules with doing things with modules when there wasn't a specific URL to work with like you have in a component.
Your case to me is exactly what com_ajax was designed to help deal with. Here's some documentation on using it:
http://docs.joomla.org/Using_Joomla_Ajax_Interface
What you're going to end up doing with your module is writing a which will get executed via the submit URL. I think you'll want to use some javascript to assist with the form submit there but I'm not sure if it's completely necessary (but it is sexier).
Then in your helper you'll set up the class for the module:
https://github.com/Joomla-Ajax-Interface/Ajax-Session-Module/blob/master/helper.php#L12
You will set up with getAjax() method:
https://github.com/Joomla-Ajax-Interface/Ajax-Session-Module/blob/master/helper.php#L12
To handle pulling in the form submit data via JFactory::getApplication()->input()
You will then return some sort of response, probably something like a JSON string which is a success or error message.
Hopefully this helps!