We report on the effect of magnetic impurities on the microscopic superconducting (SC) properties of the kagome-lattice superconductor La(Ru_1-xFe_x)₃Si₂ using muon spin relaxation/rotation. A strong suppression of the superconducting critical temperature T_c, the SC volume fraction, and the superfluid density was observed. We further find a correlation between the superfluid density and T_c which is considered a hallmark feature of unconventional superconductivity. Most remarkably, measurements of the temperature-dependent magnetic penetration depth λ reveal a change in the low-temperature behavior from exponential saturation to a linear increase, which indicates that Fe doping introduces nodes in the superconducting gap structure at concentrations as low as x=0.015. Our results point to a rare example of unconventional superconductivity in the correlated kagome lattice and accessible tunability of the superconducting gap structure, offering new insights into the microscopic mechanisms involved in superconducting order.