Releasing a new WordPress plugin based on an earlier one I wrote.
Download Forms: 3rdparty Integration. As copied from the readme:
Forms: 3rdparty Integration
Send Contact Form 7 or Gravity Forms Submissions to a 3rd-party Service, like a CRM. Multiple configurable services, custom field mapping. Provides hooks and filters for pre/post processing of results. Allows you to send separate emails, or attach additional results to existing emails. Comes with a couple examples of hooks for common CRMs (listrak, mailchimp, salesforce).
The plugin essentially makes a remote request (POST) to a service URL, passing along remapped form submission values.