While installing WebSphere application server (WAS), WebSphere Java SDK bundled with WAS is installed at path C:\Program Files\IBM\WebSphere\AppServer\java\bin
Similarly, after installing another product, example: IBM Security Directory Server [SDS] IBM Java SDK is installed at C:\Program Files\IBM\ldap\V6.4\java\bin (This may be at different level depending on how it is bundled with specific software)
After installing WAS and SDS, value(s) in PATH environment variable is not updated with reference pointing to bin location of java installed with either of these two applications. However, both work with respective installed version of IBM Java.
Is it possible for an application to run if java is installed within its installation directory without any reference pointing to bin location of java in PATH environment variable?
In a default scenario third party java application runs only if it finds (Oracle) Java installed and has following value (C:\ProgramData\Oracle\Java\javapath;) in PATH environment variable.
Does Oracle java need to be installed to run a third-party java application? Or updating PATH environment variable pointing to bin location of java installed with any of these two application would work?
Yes, it's possible and commonplace for a program to bundle Java and make use of it without adding it your systems PATH environment variable.
It's unwise to point some third-party software at a java bundled with some other application on anything but your own workstation in a pinch. You'll never remember the dependency, the java could be customized, etc.