Whenever we get a call on Avaya, I need Avaya or 3rd party App to call a URL / API and pass the CALLER_ID AND EXT_ID
to that API. e.g,
myCRM.com/{CALLER_ID}/{EXT_ID}
myCRM.com/01132899470/777
You did not reveal too much detail about your setup.
If you have AES server then you can write a TSAPI/JTAPI/DMCC client application that can monitor the stations/VDNs and call the URL with relevant parameters wherever it is appropriate for you (DELIVERED, ESTABLISHED events).
You can read about AES if you are not familiar with it: https://www.devconnectprogram.com/site/global/products_resources/avaya_aura_application_enablement_services/overview/index.gsp
Here you can read about the interfaces AES provide: https://www.devconnectprogram.com/site/global/products_resources/avaya_aura_application_enablement_services/interfaces/index.gsp