Over the last decade, rare-earth elements (REEs) have become critical in the European Union (EU) in terms of supply risk, and they remain critical to this day. End-of-life electronic scrap (e-scrap) recycling can provide a partial solution to the supply of REEs in the EU. One such product is end-of-life neodymium (NdFeB) magnets, which can be a feasible source of Nd, Dy, and Pr. REEs are normally leached out of NdFeB magnet waste using strong mineral acids, which can have an adverse impact on the environment in case of accidental release. Organic acids can be a solution to this problem due to easier handling, degradability, and less poisonous gas evolution during leaching. However, the literature on leaching NdFeB magnets waste with organic acids is very scarce and poorly investigated. This paper investigates the recovery of Nd, Pr, and Dy from NdFeB magnets waste powder using leaching and solvent extraction. The goal was to determine potential selectivity between the recovery of REEs and other impurities in the material. Citric acid and acetic acid were used as leaching agents, while di-(2-ethylhexyl) phosphoric acid (D2EHPA) was used for preliminary solvent extraction tests. The highest leaching efficiencies were achieved with 1 mol/L citric acid (where almost 100% of the REEs were leached after 24 h) and 1 mol/L acetic acid (where >95% of the REEs were leached). Fe and Co—two major impurities—were co-leached into the solution, and no leaching selectivity was achieved between the impurities and the REEs. The solvent extraction experiments with D2EHPA in Solvent 70 on 1 mol/L leachates of both acetic acid and citric acid showed much higher affinity for Nd than Fe, with better extraction properties observed in acetic acid leachate. The results showed that acetic acid and citric acid are feasible for the recovery of REEs out of NdFeB waste under certain conditions.