Specialized Use for Contact Form

I need to allow website visitors to enter one or more email addresses into a form (w/Captcha) that will generate an anonymous health alert form email with instructions. I don’t see this as an option on the most widely used contact form plugins.

Can anyone point me in the right direction, or is this likely something that will require a custom solution?

Would sending a form email like this have to involve a third party service like Mail Chimp?

Thanks!