The local density of states of a superconducting vortex core has been measured as a function of disorder in the alloy system Nb1-xTaxSe2 using a low-temperature scanning tunneling microscope. The peak observed in the zero-bias conductance at a vortex center is found to be very sensitive to disorder. As the mean free path is decreased by substitutional alloying the peak gradually disappears and for x = 0.2 the density of states in the vortex center is found to be equal to that in the normal state. The vortex-core spectra hence may provide a sensitive measure of the quasiparticle scattering time.