I can not log into the web app (portal) as an exhibitor.

Only the Primary user of the portal and the user with Admin permission can have access to the web app portal and can export the leads from the web portal.
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​Here are the steps to export leads from the web app: https://intercom.help/leadliaison/en/articles/8379694-where-can-i-export-leads-after-the-event-from-my-portal
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Please note that only the Primary user and the user with Admin permission can export leads.