Mailchimp intergration

There is a premium extension available for MailChimp integration. That’s the simplest solution:
https://rocketgeek.com/plugins/wp-members/extensions/mailchimp-integration/

There is a tutorial post in the premium support subscriber articles that describes how to integrate MailChimp into the registration:
https://rocketgeek.com/tips-and-tricks/simple-mailchimp-integration/

If you use the wpmem_post_register_data action which fires after registration (and probably the wpmem_post_update_data which is the equivalent when the user updates their profile), you could essentially integrate a MailChimp signup into your setup. You’d just need to hook a MailChimp API call to that action.
https://rocketgeek.com/plugins/wp-members/docs/filter-hooks/wpmem_post_register_data/
https://rocketgeek.com/plugins/wp-members/docs/filter-hooks/wpmem_post_update_data/