Is it possible to obtain an STC for single-pilot ops in a jet certified for two pilots?
Cessna did this sort of thing for the CE-525 series (CJ) jets. Effectively the required crew varies based on cockpit arrangement/equipment. As per the type cert 


  Minimum Crew for all Flights (see NOTE 5 for cockpit equipment/arrangement restrictions):
  
  One pilot (in the left  for all Flights pilot seat) plus additional
  equipment as specified in the Kinds of Operations Equipment List
  (KOEL) contained in the Limitations Section of the FAA Approved
  Airplane Flight Manual (AFM) 
  
  OR 
  
  One pilot and one copilot


Note 5 for reference: 


  Approval for operation with a minimum crew of one pilot is based upon
  the cockpit equipment installation and arrangement evaluated during
  FAA certification testing. No significant changes may be made to the
  installed cockpit equipment or arrangement (EFIS, autopilot, avionics,
  etc.), except as permitted by the approved MMEL, without prior
  approval from the responsible Aircraft Certification Office.


However that is the original type cert and it was done by the manufacturer, who has the resources to test these kinds of things. Doing it on your own is bound to be cost prohibitive, even considering you may have to pay a second pilot to fly with you for the entire ownership of the aircraft. 

This is also fairly similar to the case in the heavy metal world with the dropping of flight engineers and navigators when control was consolidated to the two pilots. So far as I know none of this was done via STC but appeared on the TC itself. 

Technically it may be possible, practically, it seems unlikely.

If your looking for single pilot jet performance there are plenty of early era ex-military jets hitting the market that are far faster than any business jet you can buy...