We present the first theoretical study of transport properties of Weyl semimetals with point defects. Focusing on a class of time-reversal symmetric Weyl lattice models, we show that dilute lattice vacancies induce a finite density of quasilocalized states at and near the nodal energy, causing strong modifications to the low-energy spectrum. This generates novel transport effects, namely, (i) an oscillatory behavior of the dc conductivity with the charge carrier density in the absence of magnetic fields, and (ii) a plateau-shaped dissipative optical response for photon frequencies below the interband threshold, E_{F}≲ℏω≲2E_{F}. Our results provide a path to engineer unconventional quantum transport effects in Weyl semimetals by means of common point defects.