You're correct, you will need to create an application (or service) with UCMA. This application can authenticate itself with the Lync server, create an endpoint, and then is able to create conversations/calls/messages (amongst other thing). This does not require a lync client install on the server.
Here is an MSDN video on creating a "Hello World" UCMA example application: UCMA Hello World - Send an Instant Message. I know this is about IM and not an Audio call, but once you get the IM bit, making it create an audio call instead is trivial.