Can a pilot use their GPS/FMS to fly a VOR or NDB approach?
In the US they can (Effective: May 26, 2016). 
This change allows for the use of a suitable RNAV system as a means to navigate on the final approach segment of an instrument approach procedure (IAP) based on a VOR, TACAN, or NDB signal. The underlying NAVAID must be operational and monitored for the final segment course alignment.

Still can’t fly an ILS or localizer approaches.

Update: 2017-04-19 Max Trescott has a good explanation of how to use GPS in conjunction with ground-based navaids on ILS and VOR approaches.


  AIM Section 2. Performance−Based Navigation (PBN) and Area Navigation
  (RNAV) 
  
  1−2−3. Use of Suitable Area Navigation (RNAV) Systems on Conventional
  Procedures and Routes
  
  
  Use of a suitable RNAV system as an Alternate Means of Navigation when a VOR, DME, VORTAC, VOR/DME, TACAN, NDB, or compass locator
  facility including locator outer marker and locator middle marker is
  operational and the respective aircraft is equipped with operational
  navigation equipment that is compatible with conventional navaids. For
  example, if equipped with a suitable RNAV system, a pilot may fly a
  procedure or route based on operational VOR using that RNAV system
  without monitoring the VOR.
  
  
  NOTE−
  4. The navigation database should be current for the duration of the flight.
  
  c. Uses of Suitable RNAV Systems. Subject to the operating
  requirements, operators may use a suitable RNAV system in the
  following ways.
    1. Determine aircraft position relative to, or distance from a VOR (“VOR” includes VOR, VOR/DME, and VORTAC facilities and “compass
  locator” includes locator outer marker and locator middle marker.),
  TACAN, NDB, compass locator, DME fix; or a named fix defined by a VOR
  radial, TACAN course, NDB bearing, or compass locator bearing
  intersecting a VOR or localizer course.
  
  
  Navigate to or from a VOR, TACAN, NDB, or compass locator.
  Hold over a VOR, TACAN, NDB, compass locator, or DME fix.
  Fly an arc based upon DME.
  


None of the substitutions mentioned above include the localizer portion of an ILS. So you may not substitute GPS/WAAS for and ILS or localizer approach, but you may still use it for situational awareness.

You need to dig into the AIM to find out what they mean by “suitable RNAV system” but basically both certified GPS or WAAS count.