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A stochastic algorithm is proposed, analyzed, and tested experimentally for solving continuous optimization problems with nonlinear equality constraints. It is assumed that constraint function and derivative values can be computed but that only stochastic approximations are available for the objective function and its derivatives. The algorithm is of the sequential quadratic optimization variety. Distinguishing features of the algorithm are that it only employs stochastic objective gradient estimates that satisfy a relatively weak set of assumptions (while using neither objective function values nor estimates of them) and that it allows inexact subproblem solutions to be employed, the latter of which is particularly useful in large-scale settings when the matrices defining the subproblems are too large to form and/or factorize. Conditions are imposed on the inexact subproblem solutions that account for the fact that only stochastic objective gradient estimates are employed. Convergence results are established for the method. Numerical experiments show that the proposed method vastly outperforms a stochastic subgradient method and can outperform an alternative sequential quadratic programming algorithm that employs highly accurate subproblem solutions in every iteration. Funding: This material is based upon work supported by the National Science Foundation [Awards CCF-1740796 and CCF-2139735] and the Office of Naval Research [Award N00014-21-1-2532].