Which plugin you’re using for the contact form? It looks likeyou’re thinking about Gravity forms. You can create it with a custom function when the action gform_after_submission is fired – https://www.gravityhelp.com/documentation/article/gform_after_submission/
For Contact form 7, there is another hook, see here: wpcf7_before_send_mail, wpcf7_mail_sent
http://hookr.io/plugins/contact-form-7/4.3/
Ninjaforms…http://hookr.io/plugins/ninja-forms/2.9.28/#index=a
for the other plugins you should have a look yourself.
If it’s too complicated, go for plugin solution. I have found this one for Gravity forms but haven’t tried it personally. https://wordpress.org/plugins/gravity-forms-custom-post-types/