Is there a way to approve leave so there is an audit trail?

I am sorry, but I do not understand your question. Please explain in more details what are you trying to achieve and provide us a link of a form you are working on. 

Here is the link to the form:

https://form.jotform.co/82176564703864

Once the form is completed by the employee, an email is sent to their supervisor showing a new form completed.  Is there a way in the email that gets sent to the supervisor that we can add a button in there that can either approve or decline the leave?  So we have an audit trail of who the leave request is from and who approved it?

This is very much possible to achieve using the form. Please refer to the user guide also.

User guide: How to assign a status to submissions

A more comprehensive method on how to have an approval workflow form is described here https://www.jotform.com/help/432-A-General-Approach-on-Building-an-Approval-Process-Workflow

I hope this help. Let us know if you need further assistance.