My Orion 5 string has developed quite the annoying buzz when plugged into an amp. It's definitely a ground problem, because it stops when I touch the jack or shielding paint in the electronics cavity.
 
I've tried using metal tape to ground the circuitry, but that hasn't helped.
 
Any suggestions?

First, please confirm that the annoying buzz wasn't there before and it's developed sometime after you acquired the bass.  
 
When you touch the jack or the shielding paint, are you touching them separately or at the same time?  
 
One thing to try is to make sure the nut holding the output jack in place is securely fastened. Then you can check the other nuts are secure.  
 
Let me know how this goes.

Ah! Problem solved!
 
What had happened is that one of the pickup contacts had broken loose from the wire, and I had to remove the leads from their clips and resolder. Well, I had put the wires back into the clips with the polarity reversed. BZZZZZ! Once I switched the leads around the buzzing went away.
 
FYI, the buzzing would stop when I touched the jack, the shielding paint, or the pots. It was puzzling because I had tested all the grounds with my meter and they were all showing good contact.  

That is so funny - my dad said, Wow, that sounds like one of the pickups is reversed, but we would have noticed that before shipping. I'm glad you got the problem resolved and thanks for updating me.