  We analyze the MeV/GeV emission from four bright Gamma-Ray Bursts (GRBs)
observed by the Fermi-Large Area Telescope to produce robust, stringent
constraints on a dependence of the speed of light in vacuo on the photon energy
(vacuum dispersion), a form of Lorentz invariance violation (LIV) allowed by
some Quantum Gravity (QG) theories. First, we use three different and
complementary techniques to constrain the total degree of dispersion observed
in the data. Additionally, using a maximally conservative set of assumptions on
possible source-intrinsic spectral-evolution effects, we constrain any vacuum
dispersion solely attributed to LIV. We then derive limits on the "QG energy
scale" (the energy scale that LIV-inducing QG effects become important, E_QG)
and the coefficients of the Standard Model Extension. For the subluminal case
(where high energy photons propagate more slowly than lower energy photons) and
without taking into account any source-intrinsic dispersion, our most stringent
limits (at 95% CL) are obtained from GRB090510 and are E_{QG,1}>7.6 times the
Planck energy (E_Pl) and E_{QG,2}>1.3 x 10^11 GeV for linear and quadratic
leading order LIV-induced vacuum dispersion, respectively. These limits improve
the latest constraints by Fermi and H.E.S.S. by a factor of ~2. Our results
disfavor any class of models requiring E_{QG,1} \lesssim E_Pl.
