We conducted a search for an exotic spin- and velocity-dependent interaction for polarized electrons with an experimental approach based on a high-sensitivity spin-exchange relaxation-free (SERF) magnetometer, which serves as both a source of polarized electrons and a magnetic-field sensor. The experiment aims to sensitively detect magnetic-fieldlike effects from the exotic interaction between the polarized electrons in a SERF vapor cell and unpolarized nucleons of a closely located solid-state mass. We report experimental results on the interaction with 82 h of data averaging, which sets an experimental limit on the coupling strength around 10^{-19} for the axion mass m_{a}≲10^{-3} eV, within the important axion window.