You can't run that code outside of Joomla. CB is a Joomla extension so it requires the Joomla API to function. That code is meant to be used within Joomla (e.g. 3rd party plugins).
For what you're trying to do use CB Auto Actions and its Registration action type while leaving Triggers set to None. You can then use the URL of the auto action under the Global tab to execute it directly, which you'd use in your ajax call.
To map input data to fields in this action just use [post_VARIABLE] (e.g. [post_username]) for POST or [get_variable] (e.g. [get_username]) for GET requests. It's recommended to toggle Format Functions on under the Parameters tab and sanitize your input data as much as possible. Example as follows.
Username: [cb:parse function="clean" method="string"][post_username][/cb:parse]
Email: [cb:parse function="clean" method="string"][post_email][/cb:parse]
This ensues an HTML free string only username and email address are passed to user storage.
Kyle (Krileon)
Community Builder Team Member
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If you are a Professional, Developer, or CB Paid Subscriptions subscriber and have a support issue please always post in your respective support forums for best results!
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My role here is to provide guidance and assistance. I cannot provide custom code for each custom requirement. Please do not inquire me about custom development.