One proves the existence and uniqueness in $(L^p(\mathbb{R}^3))^3$, $\frac{3}{2}<p<2$, of a global mild solution to random vorticity equations associated to stochastic $3D$ Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato [15] and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in $(L^3\cap L^{\frac{3p}{4p-6}})^3$ with respect to the time variable. Furthermore, we obtain the pathwise continuous dependence of solutions with respect to the initial data.