Do I understand correctly, that you want a 3rd party web server to show a listing of all Prompts within the UCCX Prompt Repository, and also provide hyperlinks to access the file without the user providing credentials to do so?
If so, you would need server side scripting to perform the following tasks:
1) List the folders and files in the root of the prompt repository with an HTTP GET
https://{{UCCX}}/adminapi/prompt
2) List the folders and files in a specific folder with an HTTP GET
https://{{UCCX}}/adminapi/prompt/{{folder}}
3) Retrieve a specific file with an HTTP GET
https://{{UCCX}}/adminapi/prompt/download/{{folder}}/{{file}}
How that all works on the server side, is dependent on what type of web server, the server side language you use, and the client side languages used as well.