As far as I'm aware, once the airport has installed and certified a Cat III ILS system, (which would include all the standby power and runway lighting requirements), the actual minima to be applied depend on the aircraft and operator equipment, training and certification.
The certification will be issued both by the national aviation authority of the operator and also that of the state in which the airport is situated.
So, with a Cat III approach available, the typical options may be:
Cat IIIA - DH 50' and RVR of 250m
Cat IIIB - DH 12' and RVR of 100m
Cat IIIB (no DH) with RVR of 100m
Cat IIIC (no DH or RVR)
The above are generic examples. The actual figures will depend on aircraft type and certification by the one or two states as mentioned above.
The only time I have heard of Cat IIIC being used was with the BA TriStars, but even then I think it was a theoretical capability rather than an actual operational one. I believe that they were still required to have an RVR of 50m to facilitate taxiing after landing and also to enable emergency vehicles to find them in case of a problem.
So to answer the question, you may not find any airports with Cat IIIC approaches, but there may be operators who are able to use Cat IIIC minima at certain airports.