Can I restrict access to the forums from the public?
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Yes, that's possible with Champ.
The various ways to restrict access to the forums can be found in our documentation: https://getchamp.net/guides/forums/
-- Addendum
Just to be clear. Champ does not handle any charges (ie. charging your customers). To handle charging your customers, you should stick with Shopify's own order system and manually add permissions to users for access to the forums.