Satellites in low earth orbit are moving close to 8km/sec. Most consumer-grade GPS chips still invoke the CoCom limits of 1000 knots, about 514 m/s. CoCom limits are voluntary limits for exports that you can read more about in this question and answer and this question and answer and elsewhere.
For this question, let's assume they are numerical limits in the output section of the firmware. The chip must actually calculate the speed (and altitude) before it can decide if the limit is exceeded, and then either present the solution to the output, or block it.
At 8000 m/s the doppler shift at 2GHz is about 0.05 MHz, a small fraction of the natural width of the signal due to its modulation.
There are several companies that sell GPS units for cubesats, and they are expensive (hundreds to thousands of dollars) and probably worth every penny because (at least some of them) are designed for satellite applications and space tested.
Ignoring the implementation of the CoCom limits, and all other issues of operation in space besides velocity, are there any reasons why a modern GPS chip specked at 500 m/s max velocity would not be able to work at 8000 m/s? If so, what are they?
note: 8000m/s divided by c (3E+08m/s) gives about 27ppm expansion/compression of the received sequences. This might affect some implementations of correlation (both in hardware and software).