Is it possible to have some users on a site pay by credit card and other users check out without paying?

It is possible to have two payment methods available on a site simultaneously.  The users would need to choose the correct payment method at check out.  If you need to restrict a payment method to only be available for certain users, then you would group your users into "Site Groups" and then select the groups that can see each payment method. See this page for more information:  Select here for more detail.