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X-ray absorption measurements at the U LIII, Ru K, and Fe K edges are reported for the hidden order (HO) material URu2-xFexSi2 (x=0, 0.05, 0.08, 0.10, 0.12, 0.15, and 0.20) as a function of x and temperature T. When Fe is substituted for Ru, the local structure about Fe shrinks slightly and the first neighbor Fe-Si bond length decreases by ≈0.05Å. More importantly excess disorder is observed below 80-100 K (the coherence temperature T∗) in plots of the Debye-Waller factor σ2 (σ is the width of the pair distribution function); at low T the data deviate from the usual Einstein or correlated-Debye model plots. This excess disorder is most prominent for the Ru-Si bond, and σ2 actually increases below 80 K. These results suggest a local orthorhombic distortion with B1g-like symmetry that develops below 80-100 K. A model that describes these local distortions is presented, and discussed in terms of other measurements that indicate a breaking of fourfold symmetry at low T. In addition, the square root of the difference between σ2(T) for the Ru-Si pair and a Debye fit to these data serves as an order parameter for this orthorhombic distortion, in the temperature range below 100 K. This quantity is a length related to a-b, the difference between the a and b lattice constants in the orthorhombic phase, and provides a connection between this distortion and T∗. X-ray absorption near edge structure (XANES) measurements also show that there are no changes in the edge positions down to 0.1 eV for any edge as a function of x, for T in the HO regime.