How would I set it up on a site so people CANNOT check their email via any 3rd party(e.g. Outlook, Thunderbird, Etc.) and ONLY be able to check via the webmail I provide? then If I decide I can allow certain people the ability to check their email via 3rd Party.
You can disable POP3/IMAP for accounts using Zimbra.
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You may turn off the standard POP3 (110) and IMAP (143) ports , then change them with custom ports you want to use for your 'WEBMAIL' application (Most WebMail apps use IMAP to connect) - you will also need to configure your webmail to connect to that custom IMAP port.
You may also give your custom POP3 ports to your selected clients only, so that they may use it with 3rd party software, such as: outlook, thunderbird, etc.