Can a private pilot get an RVSM LOA?
An RVSM LOA is given to an operator not a pilot. 

RVSM airspace can be used under part 91, so it is not limited to commercial operations. It can be a private owner or a non-aviation business. FAA Advisory Circular 91-85A says this:


  3.4.1 Who is the Correct Person to Apply for and Have Issued the RVSM Authorization?
  The person exercising operational control of the aircraft during the operation requiring an RVSM authorization is the proper person to be the applicant for that authorization. It is important to note it is the RVSM applicant’s responsibility to submit a request for RVSM authorization in the name of the person having operational control of the aircraft, not the responsibility of the FSDO or a specific ASI to make such a determination. The following general information may be useful in assisting the RVSM applicant in determining if the appropriate party has been properly designated as the legal Operator with respect to the RVSM authorization request:
  
  
  For non-commercial operations conducted under part 91 and part 125
  (A125 LODA holders), the authorization applicant and legal Operator should normally be one of the following persons, in which event the authorization will be issued in the form of an appropriate letter of authorization (LOA):
  
  
  A registered owner of the aircraft operating the aircraft incidental to its own non-air transportation business or personal activity.
  A person assuming operational control of the aircraft through a lease or use agreement for that person’s operation of the aircraft incidental to that person’s own non-air transportation business or personal activity.
  
  
  
  *emphasis mine


So if it is a company plane then the company will be the holder of the RVSM authorization, and there must be a responsible party within the company that is knowledgeable about the operation of the aircraft (i.e. responsible to ensure maintenance requirements are met, the pilots are “RVSM-knowledgeable,” that proper procedures are followed, etc. - not necessarily a pilot).

The authorization will be specific to the aircraft (or a group of similar aircraft) and it must meet the equipment requirements laid out in Part 91 apppendix G.

So, what makes the pilot “RVSM-knowledgeable?”

The same advisory circular says:


  3.3.1.1 For an applicant operating only under part 91 or part 125 (including
  A125 Letter of Deviation Authority (LODA) holders), demonstrating it has RVSM-Knowledgeable Pilots will consist of providing sufficient evidence each pilot has an adequate knowledge of RVSM requirements, policies, and procedures as required in part 91 appendix G, section 3(c)(2).
  
  
  The following are acceptable means for the Operator to show the FAA its pilots have adequate knowledge of the RVSM operating practices and procedures: 14 CFR part 142 training center certificates without further evaluation; certificates documenting completion of a course of instruction on RVSM policy and procedures; and/or an Operator’s in-house training program.
  
  
  Note: The FAA, at its discretion, may evaluate a training course prior to accepting a training certificate.
  
  *emphasis mine


Most likely, in a non-aviation company there won’t be an acceptable in-house training program, so any pilots will need certificates documenting completion of a course of instruction on RVSM policy and procedures. There are several flight schools with approved courses that can provide such certificates.

What it boils down to is the company that owns or leases the plane must get an LOA for their aircraft. The aircraft must meet RVSM requirements and to use RVSM airspace the pilot also must be certified. 

If you own your own aircraft then you are, by default, the operator. So you can apply for an LOA as the operator. You would be your own responsible party and anyone who flies the aircraft in RVSM airspace, be that yourself or someone else, must be an “RVSM-knowledgeable” pilot.

If you rented your personal plane out to someone, say to a flight school, then they would become the operator while it is in their possession, so they would need to have their own LOA to operate in RVSM airspace.