The composition of planets is largely determined by the chemical and dynamical evolution of the disk during planetesimal formation and growth. To predict the diversity of exoplanet compositions, previous works modeled planetesimal composition as the equilibrium chemical composition of a protoplanetary disk at a single time. However, planetesimals form over an extended period of time, during which elements sequentially condense out of the gas as the disk cools and are accreted onto planetesimals. To account for the evolution of the disk during planetesimal formation, we couple models of disk chemistry and dynamics with a prescription for planetesimal formation. We then follow the growth of these planetesimals into terrestrial planets with N-body simulations of late-stage planet formation to evaluate the effect of sequential condensation on the bulk composition of planets. We find that our model produces results similar to those of earlier models for disks with C/O ratios close to the solar value (0.54). However, in disks with C/O ratios greater than 0.8, carbon-rich planetesimals form throughout a much larger radial range of the disk. Furthermore, our model produces carbon-rich planetesimals in disks with C/O ratios as low as ~0.65, which is not possible in the static equilibrium chemistry case. These results suggest that (1) there may be a large population of short-period carbon-rich planets around moderately carbon-enhanced stars (0.65 < C/O < 0.8) and (2) carbon-rich planets can form throughout the terrestrial planet region around carbon-rich stars (C/O > 0.8).