Can we extract the Primary Key of the Table Using Web Services API?

No is the short answer.

The OBIEE SOAP WSDL API (web service) does not provide logical or physical layer metadata information.
It is a presentation services / subject area based API.

The only other "service" that you can use to find this would be to use the bijdbc.jar library which you add to a Java Library / application. From there you can pull back information regarding table metadata, etc.

Depending on your implementation the other option would be to extract the metadata manually from the RPD using several commands possible there.