Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. In this paper, we numerically demonstrate that the normalized localization length $\beta$ of the eigenfunctions of multilayer random networks follows a simple scaling law given by $\beta=x^*/(1+x^*)$, with $x^* = \gamma(b_{eff}^2/L)^\delta$, $\gamma,\delta\sim 1$ and $b_{eff}$ being the effective bandwidth of the adjacency matrix of the network, whose size is $L=M\times N$. The reported scaling law for $\beta$ might help to better understand criticality in multilayer networks as well as used as to predict the eigenfunction localization properties of them.