We investigate the induced orbital magnetization density in a Rashba electron gas with magnetic impurities. Relying on classical electrodynamics, we obtain this quantity through the bound currents composed of paramagnetic and diamagneticlike contributions which emerge from the spin-orbit interaction. Similar to Friedel charge ripples, the bound currents and the orbital magnetization density oscillate as a function of distance away from the impurity with characteristic wavelengths defined by the Fermi energy and the strength of the Rashba spin-orbit interaction. The net induced orbital magnetization was found to be of the order of magnitude of its spin counterpart. In addition to the exploration of the impact of the electronic filling of the impurity states, we investigate and analyze the orbital magnetization induced by an equilateral frustrated trimer in various noncollinear magnetic states. On the one hand, we confirm that nonvanishing three-spin chiralities generate a finite orbital magnetization density. On the other hand, higher-order contributions lead to multiple-spin chiralities affecting nontrivially and significantly the overall magnitude and sign of the orbital magnetization. This study substantiates the notion that the orbital degrees of freedom are an essential aspect of nanoscale magnetism, calling for further theoretical and experimental attention.