The form tag does not need an action and not having one will not prevent a form from submitting.
There are two other possibilities to check.
Make sure that your them has wp_footer(); in the correct place
The other thing that can cause the submission to fail is a JavaScript Error or a PHP error during the AJAX call. Check that there are no JS errors and turn on error debugging and logging to see if any PHP errors are occurring during the AJAX call. https://codex.wordpress.org/WP_DEBUG
