How can I check that the particular version of chef-client is having this much of gems which was different from previous version of chef client.
I can see that previous versions of chef client were not having the chef-vault gem but this chef-client latest version 13.8.6 which i installed is having chef-client gem in it.
So just wanted to be sure that if i install chef client 13.8.6 version then do i need to write explicitly any code to install the gem through cookbook or the chef-client installation will itself handle the gem of chef vault
The chef-vault gem is included in chef-client as of version 13.4, so you are correct that any node running that version or later does not require any code to install the gem before using it (although in my experience, nothing bad happened by including it again).
The best way to check for these kinds of changes is to read all the release notes from the version you were running to the version to which you are upgrading: https://docs.chef.io/release_notes.html#what-s-new-in-13-4-19
You can also check the changelog in the chef-client github repository for sometimes slightly more details: https://github.com/chef/chef/blob/master/CHANGELOG.md