OpenJDK is basically the Java JDK. Oracle JDK is only a “fork” of it, but they have complete feature-parity. Oracle only provides paid support for current and older versions of OpenJDK (similarly to many other vendors)

So you can just install OpenJDK from a package manager, use the jlink tool and create your own small “JRE” that you want to bundle. Jpackage can create a platform native executable from that, that is either an installable exe, .deb, rpm or MacOS’s format (I forgot it’s name)