A polymer growth Hamiltonian with an accompanying novel lattice has been constructed to model reaction dynamics of polydisperse polymer systems that have kinetics affected by an environment which is altered over time by the polymerization process itself. Monte Carlo dynamics are used to simulate the polymerization process with bond breaking/formation moves analogous to a correlated percolation model. Lattice events thus occur on the time scale of reactivity rather than the much shorter time scales of either monomer motion or conformational dynamics, though an effort is made to include the latter as well. The equilibrium behavior of the system has been well mapped out and is complex enough to model a variety of real polymer systems such as solid state polymerization, living polymers and thermosetting polymers. Detailed equilibrium structural information such as molecular weight distributions and extent of cross polymerization have been obtained. Pseudodynamic information such as the time-dependence in the polymer weight distribution and more detailed quantities may also be obtained with respect to Monte Carlo time scales.