We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where in each iteration a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to employ indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm needs to address an infeasibility issue – the linearized equality constraints and trust-region constraints might lead to infeasible SQP subproblems. In this regard, we propose an textitadaptive relaxation technique to compute the trial step that consists of a normal step and a tangential step. To control the lengths of the two steps, we adaptively decompose the trust-region radius into two segments based on the proportions of the feasibility and optimality residuals to the full KKT residual. The normal step has a closed form, while the tangential step is solved from a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish the global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.