so you are looking to connect your application server to a 3rd party application server?
I would suggest you establish what Auth system your 3rd party uses. Likely that they would be implementing at least a token authentication. They give you a token and you pass it in a header with your requests to their server to authenticate.
More complicated but still supported with be to connect via OAuth https://www.varonis.com/blog/what-is-oauth but you'll need to be lead by their existing authentication.