The dynamics of an elastic network model for the enzyme 4-oxalocrotonate tautomerase is studied in a system crowded by mobile macromolecules, also modeled by elastic networks. The system includes a large number of solvent molecules, as well as substrate and product molecules which undergo catalytic reactions with this hexameric protein. The time evolution of the entire system takes place through a hybrid dynamics that combines molecular dynamics for solute species and multiparticle collision dynamics for the solvent. It is shown that crowding leads to subdiffusive dynamics for the protein, in accord with many studies of diffusion in crowded environments, and increases orientational relaxation times. The enzyme reaction kinetics is also modified by crowding. The effective Michaelis constant decreases with crowding volume fraction, and this decrease is attributed to excluded volume effects, which dominate over effects due to reduced substrate diffusion that would cause the Michaelis constant to increase.