Polymerlike structures are ubiquitous in nature and synthetic materials. Their configurational and migration properties are often affected by crowded environments leading to nonthermal fluctuations. Here, we study an ideal Rouse chain in contact with a nonhomogeneous active bath, characterized by the presence of active self-propelled agents which exert time-correlated forces on the chain. By means of a coarse-graining procedure, we derive an effective evolution for the center of mass of the chain and show its tendency to migrate toward and preferentially localize in regions of high and low bath activity depending on the model parameters. In particular, we demonstrate that an active bath with nonuniform activity can be used to separate efficiently polymeric species with different lengths and/or connectivity.