Can a standard user upgrade OSX versions or does it have to be an admin user? I administer several macOS computers, which are a mixture of macOS 10.11.6 and macOS 10.13.6. 

10.13.6 is the current maximum version of macOS we can go to without getting software compatibility issues. 

I have a local admin account on each computer and the individual users are setup with standard user privileges. 

Ive noticed on the standard user accounts they keep getting a notification to upgrade to Mojave 10.14. I think one of them clicked this by mistake as in their applications folder they had the Mojave installer. 

Is it possible for a standard user to start the upgrade or will they be blocked by requiring an administrators password ?
In order to make any "deep" changes to macOS it will ask for an Administrator username and password.

The only way a non-admin user can get around this limitation is by knowing the username and password of an admin user.

So no they can't use that app to upgrade their systems.

They could however copy the app off and make a bootable USB installer on another Mac and then boot their work Mac from that USB to upgrade it, depending on how technical they are. or how determined.