As this is a pretty advanced topic, I will assume that you are familiar with Ajax and WP development. If you are not, please study up, or hire a developer and point them to this page.
After loading the HTML form, the necessary events need to be registered on the form. This is done by calling 2 functions:
wpcf7.init($form), which will register the Contact Form 7 events.
wpcf7cf.initForm($form), which will register the Conditional Fields for Contact Form 7 events.
note: $form should hold the jQuery form element.
const $form = jQuery('.wpcf7-form').eq(0).
The following example should get you on your way: https://conditional-fields-cf7.bdwm.be/ajax-example/