I work for Calendly. There's not an API to programmatically schedule events, and probably won't be for a while. You can review the latest API documentation and submit feedback at https://developer.calendly.com/.
UPDATE 1
While there's still not a public API for scheduling events, we now have a public API endpoint to List Available Slots for an Event Type. You can use it to deepen an integration between Calendly and your app, like presenting a custom availability schedule. However, the final booking will still have to be done through Calendly.
UPDATE 2
There are also two more API endpoints that may be helpful:
- List User Busy Times - use it to get a User's "raw" availability, outside of the scope of a particular Event Type. Returns a list of busy time slots for a given User and date range.
- List User Availability Schedules - use it to get e.g. default working hours of a User in an Organization.