I have a Wordpress page using Contact Form 7. Is there a plugin I could install that would let me subscribe emails entered into this form to my Mailchimp List without changing the design of the form? Or could I add some code in the inside functions.php, maybe to call the API? The form is at the bottom right of this page:
While you can always try and custom code the functionality which is what I'd recommend if you enjoy coding. But if you want just the solution then you can use the CF7 MailChimp Extension which, I am sure, will exceed your expectation and the integration is just perfect.