How can a part 91 operator gain authorisation from the FAA to perform RNAV (RNP AR) approaches?
I have no practical experience with FAA RNP AR authorization but the guidelines for obtaining RNP AR authorization from the FAA are described in FAA Advisory Circular 90-101A (FAA PDF). The whole process to go through is described in that document. It specifically mentions Part 91 and Part 91K (fractional ownership) so it should be possible to obtain permission for individual pilots. 

However, it will not be easy for an individual pilot to meet the requirements. 

You will have to perform a high number of RNP AR approaches under a provisional authorization in VMC or increased minima before final authorization is granted.

Luckily the FAA made provision for operators with a small fleet:


  NOTE: In unique situations where the completion of 100 successful
  approaches could take operators an unreasonably long period of time
  due to factors such as a small number of aircraft in the fleet,
  limited opportunity to use runways having appropriate procedures,
  etc., consideration of a reduction in the required number of
  approaches will be on a case-by-case basis. The FSDO/CHDO will
  coordinate with FAA HQ in consideration of an operator’s request for
  reduced interim authorization requirements.


You will need to arrange proper training:


  Initial training: With no prior RNP AR approach experience,
  each pilot
  must complete at least four RNP AR approach
  procedures: two as pilot
  flying and two as pilot
  monitoring.These four RNP AR
  approaches shall employ the uni
  que AR characteristics of the
  operator’s approved procedures (i.e., RF legs, RNP missed). You must fly two approaches to the DA and two approaches must result in an RNP missed approach procedure. Two of the above approaches will include interrupted approaches resulting in one approach with vectors to resume the approach and one approach resulting in a hold at an initial approach fix (IAF) or transition
  fix. 


And in addition to the a qualifying aircraft the operator must have a program to ensure data the navigation database stays up to date and is of sufficient quality.

You will also need to implement a monitoring program to review your RNP AR approaches and detect any negative trends. Results from this program will need to be shared with the FAA on a monthly basis.