This paper develops a distributed algorithm for data-driven Distributed Model Predictive Control (DMPC) for multi-agent control systems, where the agents' dynamics are modeled by Gaussian Processes (GPs). A multi-agent control system with a coordinator is considered, in which computation and data must be distributed among the agents and the coordinator. We employ the linearized Gaussian Process (linGP) concept, proposed in our previous works, to sequentially approximate the stochastic latent processes of the GP models in a Sequential Convex Programming (SCP) framework, leading to a convex linGP-DMPC subproblem, which is solved cooperatively by the agents with the ADMM algorithm. The resulting distributed algorithm, called linGP-SCP-ADMM, can solve nonconvex GP-DMPC for multi-agent systems effectively since the data and computation are distributed among the agents. The effectiveness and advantages of the proposed algorithm are evaluated by simulation in a formation control example.