JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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Have you tried following this guide on how to integrate MailChimp to your JotForm form: https://www.jotform.com/help/163-How-to-Integrate-with-MailChimp?
In the MailChimp integration wizard, you'll need to add your API key that you got from your MailChimp account:
If it is valid, hit the "Authenticate" button. Then complete the other settings.
Let us know if you are still unable to make this work.