Altermagnetic superconductors offer the possibility of exploring unconventional superconductivity, including topological states and finite-momentum superconductivity, with promising applications in spintronics and quantum information. However, a direct experimental confirmation of their existence remains elusive. In this work, we propose nonmagnetic impurities as probes of the interplay between altermagnetism and superconductivity. These impurities induce spin-polarized subgap states whose spatial extension reflects the magnetic order of the substrate material. Depending on whether or not the impurity respects the bulk symmetries, these states form spin-degenerate or spin-split doublets. An external magnetic field aligned with the Néel vector can further control the doublet splitting. We further demonstrate that coupling between impurity states leads to a position-dependent, spin-sensitive hybridization, enabling another approach for in situ control of atomic-size quantum devices. These findings provide unambiguous experimental signatures of altermagnetic superconductivity accessible via local measurements such as scanning tunneling microscopy and open unexplored pathways for designing tunable quantum devices.