How do get approval to integrate jotform with my Authorize.net account?

Approval is not required to use any payment integrations or any features for that matter.  Here is more information on setting up payment fields in your form:

https://www.jotform.com/help/323-Mastering-Payment-Form-Integrations-with-JotForm

The integration is setup after adding an Authorize.net payment field to your form.