The quickest way to get this rolling would be to buy a relay from relay.tools (12k sats/mo), and configure it to reject access to everyone other than the npubs you add to the access list. Then, add the npubs of all members of this group to the access list. All users will be able to connect to the relay by browsing to your relay URL, and authenticating using the Nos2x signing extension. All approved users will then be able to chat exclusively with the group right from the browser. So it's a relay and a group chat client all in one.