Thermonuclear supernovae originating from the explosion of a white dwarf accreting mass from a companion star have been suggested as a site for the production of p nuclides. Such nuclei are produced during the explosion, in layers enriched with seed nuclei coming from prior strong s processing. These seeds are transformed into proton-richer isotopes mainly by photodisintegration reactions. Several thousand trajectories from a 2D explosion model were used in a Monte Carlo approach. Temperature-dependent uncertainties were assigned individually to thousands of rates varied simultaneously in post-processing in an extended nuclear reaction network. The uncertainties in the final nuclear abundances originating from uncertainties in the astrophysical reaction rates were determined. In addition to the 35 classical p nuclides, abundance uncertainties were also determined for the radioactive nuclides 92Nb, 97, 98Tc, 146Sm, and for the abundance ratios Y(92Mo)/Y(94Mo), Y(92Nb)/Y(92Mo), Y(97Tc)/Y(98Ru), Y(98Tc)/Y(98Ru), and Y(146Sm)/Y(144Sm), important for Galactic Chemical Evolution studies. Uncertainties found were generally lower than a factor of 2, although most nucleosynthesis flows mainly involve predicted rates with larger uncertainties. The main contribution to the total uncertainties comes from a group of trajectories with high peak density originating from the interior of the exploding white dwarf. The distinction between low-density and high-density trajectories allows more general conclusions to be drawn, also applicable to other simulations of white dwarf explosions.