Using Monte Carlo simulations, we study the diffusion-controlled dynamics of a polymer chain, which leads to the intrachain reaction of two reactive monomers. On the basis of the bond fluctuation lattice model, we computed the mean first-passage time and examined its scaling behavior as a function of the chain length and location of the reactive monomers, for five different types of polymer architectures. While the simulation results confirmed previous theoretical predictions for the free chain case, our measurements of the mean first-passage time also yielded new insights into the looping dynamics of end-grafted chains.