Hi,
The old way of doing this was a bit clunky. You would use a plugin like Contact Form 7 Dynamic Text Extension in combination with a shortcode that generates the current date string. You would create a dynamic shortcode field and map it to your PDF field.
There is a new feature that will soon be released that allows you to map mail-tags to PDF fields. You can then use the ‘[_date]’ mail-tag. See more information at https://contactform7.com/special-mail-tags/. If you want to try out the new unreleased feature, download and install the work in progress code at https://github.com/maximum-software/pdf-forms-for-contact-form-7/archive/master.zip.
The new version of the plugin that has the mail-tags feature has been released.