Is it legal to fly the localizer approach when cleared for the ILS?
The question really boils down to: "ATC clears me to fly an approach.  Is the approach clearance defined by what is on the approach plate, or is it specific to the type of navigation equipment used?"  It turns out that it is based on the approach plate.

The answer is found here:
https://www.faa.gov/air_traffic/publications/atpubs/aim/aim0504.html
Section 5-4-5.a.3(a)


  More than one navigational system separated by the word “or” indicates
  either type of equipment may be used to execute the final approach
  (e.g., VOR or GPS RWY 15).


When the chart is written as in your example, you are cleared for the approach, and you must use one of the approved types of navigational equipment to fly it.  There is no requirement in FAR/AIM to tell approach control which equipment you will be using (beyond the equipment code in your flight plan).  

As further corroboration, section 5-4-7.d in the above document specifies that ATC is allowed to clear you for an approach even if the equipment required for some types of navigation systems is not available.  This actually is not that rare because NDB/GPS approaches are still used even if the NDB is not working.

The requirement to notify ATC in the event of navigation radio failure is separate.  Oddly, you don't have to tell ATC if you have a glideslope receiver at all, but you are supposed to tell them if one that you do have breaks.  If it happened while you were actually flying the approach, I would encourage you to execute a missed approach.  But if you just want to fly an ILS/LOC approach as localizer-only for practice, ATC doesn't care and you don't need to tell them.