The entire underpinnings of GPS relies on the speed of light to calculate positions as the GPS satellites send their position, velocity and a current time stamp. These can be used to generate a distance from the satellite based on the velocity of the light. Through multiple distance readings of multiple satellites a position can be determined. Relativity does in fact play a role since the satellites are not geosynchronous they are moving in comparison to Earths reference frame. This creates non negligible errors in GPS which have been accounted for in the GPS equations. If they hadn't been we would have much more error in GPS
there are at least two relativistic effects involved in gps timings - http://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf - but they would not be solved if the satellites were in geosynchronous orbit as the earth itself does not have a single non-accelerating reference frame.
