We develop approximate estimation methods for exponential random graph models (ERGMs), whose likelihood is proportional to an intractable normalizing constant. The usual approach approximates this constant with Monte Carlo simulations; however, convergence may be exponentially slow. We propose a deterministic method, based on a variational mean-field approximation of the ERGM's normalizing constant. We compute lower and upper bounds for the approximation error for any network size, adapting nonlinear large deviation results. This translates into bounds on the distance between true likelihood and mean-field likelihood. Monte Carlo simulations suggest that in practice, our deterministic method performs better than our conservative theoretical approximation bounds imply, for a large class of models.