Axions or axion-like particles (ALPs) are well-motivated dark matter (DM) candidates whose coupling to photons induces periodic oscillations in the polarization angle of astrophysical light. This work reports the first search for such a signature using ten years of optical polarimetric monitoring of the blazar 1ES~1959+650. No statistically significant periodicity is detected, determined by a Lomb-Scargle periodogram combined with Monte Carlo analysis. Assuming a central DM density in the host galaxy, this null result places tight upper limits on the ALP-photon coupling constant at $g_{a\gamma }<(5.8\times {10}^{-14}-1.8\times {10}^{-10}){GeV}^{-1}$ across a broad ALP mass range of $m_a\sim (1.4\times {10}^{-23}-5.2\times {10}^{-20})eV$. Our constraints surpass those from Very Long Baseline Array polarimetry of active galactic jets and from polarimetric data of the European Pulsar Timing Array, and are competitive with those from long-term Galactic pulsar timing of PSR J0437-4715 over the same ALP mass window. These results establish long-term blazar polarimetry as a competitive and complementary approach for probing axion-like DM on extragalactic scales.