We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.