We propose a minimal one-dimensional continuum model for the spontaneous initiation of protrusion-driven cell crawling on a rigid substrate. The cell cytoskeleton is represented as a viscous actin meshwork that turns over in the bulk and polymerizes at two moving cell edges. Symmetry breaking arises from the feedback between cell motion, an external chemical regulator of actin nucleation, and actin polymerization at the cell fronts. When the cell moves, the regulator becomes polarized around the moving boundaries, thereby imposing different actin nucleation densities at the two edges. This generates unequal protrusive rates, which in turn reinforce motion and sustain the chemical polarization. Above a critical protrusive activity, the static symmetric state loses stability and the system undergoes a bifurcation toward a motile polarized state. Depending on how the external cue controls actin nucleation, the transition can be either supercritical or subcritical, leading in the latter case to coexistence between static and motile states. Using parameter values appropriate for keratocyte cells, the model predicts crawling speeds of the correct order of magnitude and actin-density profiles that are qualitatively similar to experimental observations, including asymmetric edge-localized density peaks. These results identify a generic mechanism by which external biochemical regulation of actin nucleation can trigger spontaneous motility along a one-dimensional track without requiring molecular motors, specific adhesion dynamics, deformable substrates, or pre-existing polarity.