Ahhh Sui he can just show up to a class on campus?
It depends. I'm currently enrolled in an NJ university (not Rutgers) and we have the same requirement. However, the courses I'm taking are all "Online Only" and therefore I'm not required to show proof of vaccination, since there isn't any on-campus attendance. There are hybrid (on-campus with live video) courses though so it's possible Rutgers has something similar.
You can't just show up to any class/course in a college - you need to register.
Although, the way the (Stone Age) backend systems work at many of these universities I wouldn't be surprised if this is just a technical limitation- like Dan is saying.