This will guide you on how to search and get lead recordings through API. 

(Note: First is you need to add Authorization Token, click here to see complete details)

From the Dashboard, click on API and select Leads from the drop-down.

Click on Get Lead Recordings tab. See required fields below for you to do this. 

auth_token 
lead_id

Scroll down to the bottom to see the Example PHP Usage and the Example JSON Response.

To test this feature, go to Test Request tab and fill in the information you wanted to show on the test insert.

Select the correct Authorization Token for this API from the drop down box.

Note: If you don't know the Start/End time of the recording, you can clear the fields out.

From the sample above, I enter the lead's ID number.

Enable Show Query to get the Request Query String and click Run button.

This will show you all the recordings available for the said Lead ID.