We usually use the CFDB7 plugin to save the data. (https://es.wordpress.org/plugins/contact-form-cfdb7/). More information (https://github.com/mdsimpson/contact-form-7-to-database-extension). This is supposed to be valid for any form submission plugin.
If we want to save all the shipments that are made so we can make debug. We use the Email Log plugin (https://es.wordpress.org/plugins/email-log/)