How to call a SOAP web service from a Wordpress form?

I wish I had a better answer for you, but I'm afraid it'll involve some coding and understanding on SOAP. (Because I couldn't find any WordPress plugins that implemented a generic SOAP client or more specifically a form submitter with SOAP support).

This post author collected a nice group of links that should help get you started: https://wordpress.stackexchange.com/questions/14804/embedding-a-soap-client-into-a-wordpress-plugin

Personally, I use GravityForms for all my projects (but it's a paid plugin). But they provide a nice API and there's a hook called gform_post_submission that I've personally used a lot. (I've "posted" to REST services using that hook).

There is a soap plugin framework for wordpress that allows you to create/expose wordpress behavior as soap api. http://www.amazon.com/dp/B00D1AYIG8 and http://github.com/patrickingle/wp-soap-services