When you use the iframe, you are just displaying what the other site provides. You don't have a way of connecting this to the DNN site.
You will need to integrate the form into DotNetNuke. You can use the form, but you will need to do some modifications to the form to remove the form post and use javascript to accomplish that.
It might be easier to use something like Dynamic Forms which can do what you need all in the context of the DotNetNuke site.