It is possible but does often require a Salesforce developer who can work with you to write some code.
1. button for your external application for them to click when they receive the Payment Request
2. Automatically Create PR Record is set under Chargent Settings (Advanced Settings). This will automatically generate and log the Payment Request when an Order, Opportunity, or Case is created.
3. Create the code that will allow you to export this field and merge it into the external application
If you are looking to send individual Payment Requests and not send to every Chagent Order then you would need to create the code that will allow the above to happen as well as the option below.
4. Create the code that will write "New Payment Request" button in the Related List for Payment Requests in that object for reference.