When build on Webflow a directory site typically needs;
If you're comfortable with Javascript development, and building automations through platforms like Make, you can build this using Memberstack ( for user accounts, billing, user id, and user-specific private data storage ) and Make ( for the CMS updates of the public-facing directory ).
Alternatively you could use Wized, which offers some nocode development capabilities. You might need Xano as well for the internal user account storage and billing.