You can perform umrah on a visit visa.
By chance someone posted this very question in another forum and this is what was said.
There are checkpoints before you enter Makkah & Madinah - I drove through a few of them and wasn't stopped.
However if you get stopped and don't have a Visa, I can see you having problems!
If you're asking whether it's possible to perform Umra with a visit visa: Yes it is and most likely you won't be stopped at the check points before Makkah & Madinah.
If you're asking whether Saudi law allows you to do that: No it doesn't. And I feel you wouldn't really want to break the law while going on a spiritual journey. In the unlikely case that you will get stopped you might face problems on top of that.