We predict a supercurrent-driven N'eel spin-orbit torque in a superconductor/$d$-wave altermagnet heterostructure, associated with the emergence of spin-triplet correlations. The effect can be understood as a consequence of the supercurrent-induced spin polarization, owing to the interplay between spin-orbit coupling and momentum-dependent spin splitting, as found, for example, in altermagnets. Remarkably, the supercurrent can be tuned by the N'eel-vector direction, and the supercurrent-induced torque can both propel magnetic domain walls and reverse the N'eel-vector orientation within a domain wall. These findings establish superconductor/altermagnet heterostructures as a versatile platform for the {supercurrent-driven} control of the N'eel vector, with potential applications in racetrack memory, dissipationless superconducting electronics, and unconventional computing.