Build API without the WolframCloud?

Currently APIFunction / CloudDeploy works only with the Public Cloud or an Enterprise Private Cloud.

So either you need an EPC on your side or you can use WebMathematica or Mathematica (WebMMA license required anyway). Though WebMMA does not support APIFunctions anyway so you'd have to adapt it to create JSP based api interface.

Let's hope someone in WRI sees the potential and an upgrade for WebMMA is on the way.

Now there's WolframWebEngine which allows one to create a sort of personal wolfram cloud.