*Update:* Found the answer in this other support question: https://wordpress.org/support/topic/executing-form-requires-access-to-wp-admin?replies=4
The problem was the Ajax submission. Disabling Ajax on the form fixes the problem.
Our site has an HTACCESS security protocol for anyone trying to access admin level pages. This isn't a problem for regular users visiting the site but for some reason when a form is submitted through Easy Forms for MailChimp the HTACCESS prompt is triggered and unless admin-level credentials are submitted, the form won't submit. This doesn't happen with WPCF7 forms. Any ideas how we can get around this?
https://wordpress.org/plugins/yikes-inc-easy-mailchimp-extender/