We study the frequency dependence of the optical conductivity $text{Re} , sigma(omega)$ of the Heisenberg spin-$1/2$ chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random $z$-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform $text{Re} , sigma(omega)$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $text{Re} , sigma(omega)$ is well described by $text{Re} , sigma(omega) approx sigma_text{dc} + a , |omega|^alpha$, with $alpha approx 1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of $sigma_text{dc}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of $sigma_text{dc}$ is consistent with the existence of a mobility edge.
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