We consider the problem of max-min beamforming (MMB) for cell-free massive multi-input multi-output (MIMO) systems, where the objective is to maximize the minimum achievable rate among all users. Existing MMB methods are mainly based on deterministic optimization methods, which are computationally inefficient when the problem size grows large. To address this issue, we, in this paper, propose a randomized alternating direction method of multiplier (ADMM) algorithm for large-scale MMB problems. We first propose a novel formulation that transforms the highly challenging feasibility-checking problem into a linearly constrained optimization problem. An efficient randomized ADMM is then developed for solving the linearly constrained problem. Unlike standard ADMM, randomized ADMM only needs to solve a small number of subproblems at each iteration to ensure convergence, thus achieving a substantial complexity reduction. Our theoretical analysis reveals that the proposed algorithm exhibits an O(1/\bar{t}) convergence rate (\bar{t} represents the number of iterations), which is on the same order as its deterministic counterpart. Numerical results show that the proposed algorithm offers a significant complexity advantage over existing methods in solving the MMB problem.