Edit: I re-read the question - the below only gets you the user running the build (which technically is often more interesting), not the one triggering the build in the frontend (be it REST-API or WebUI).
If you have Jenkins impersonation enabled, then I believe the result should be equivalent, otherwise this will only get you the user who owns the jenkins agent on the build machine.
Original answer:
Another way would be to
sh 'export jenkins_user=$(whoami)'
Downside: Linux-dependent, difficult to port across multiple agents in a single build (but then, the auth context may be different on each slave)
Upside: No need to install plugins (which on shared/large Jenkins instances can be tricky)