We study the stochastic homogenization for a Cauchy problem for a firstorder Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like H(x, σ(x)p, ω) where σ(x) is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the ε-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem.