Are there experiments which show that single photons (not classical em waves) travel exactly at $c$ in vacuum? What is the error bar in that case? The question is posed due to the fact that loop quantum gravity predicts variations in the speed of light.
This link summarizes the measurements of the speed of light.
The first measurement of c that didn't make use of the heavens was by Armand Fizeau in 1849. He used a beam of light reflected from a mirror 8 km away. The beam was aimed at the teeth of a rapidly spinning wheel. The speed of the wheel was increased until its motion was such that the light's two-way passage coincided with a movement of the wheel's circumference by one tooth. This gave a value for c of 315,000 km/s. Leon Foucault improved on this result a year later using rotating mirrors, which gave the much more accurate value of 298,000 km/s. His technique was good enough to confirm that light travels slower in water than in air.
After Maxwell published his theory of electromagnetism, it became possible to calculate the speed of light indirectly by instead measuring the magnetic permeability and electric permittivity of free space. This was first done by Weber and Kohlrausch in 1857. In 1907 Rosa and Dorsey obtained 299,788 km/s in this way. It was the most accurate value at that time.
Many other methods were subsequently employed to further improve the accuracy of the measurement of c, so that it soon became necessary to correct for the refractive index of air since c is light's speed in a vacuum.
So the speed in vacuum is accessed through the refraction index corrections for air.
Classical electromagnetic beams emerge from a confluence of photons an the velocity measured is the group velocity of light.
Have a look at this answer and links therein which discusses the experiment where the relative velocity of two photons is measured to retain the slow group velocity of an optical path through a medium.
I do not know whether single photon experiments have been attempted and whether finding a constant speed c in earth experiments would say anything about the loop quantum gravity (afaik it has lorenz invariance problems too).
Later addition on excluding gravitational models:
Especially the Fermi-LAT group was able show that no energy dependence and thus no observable Lorentz violation occurs in the photon sector even beyond the Planck energy, which excludes a large class of Lorentz-violating quantum gravity models.
Lorenz violation and the constancy of speed of light in vacuum are correlated.
Also vacuum birefringence studies would show up Lorenz violations.
The very strict limits from the measurements are used to exclude alternate gravitational models, even though individual photons are not measured.