Summary: Constrained filters, through utilising the prior state constraint information, are designed to obtain more accurate state estimates in applications, and most of them deal with the estimation problem of systems with deterministic constraints. In practice, complex environmental disturbance, incomplete information or uncooperative behaviour often brings out uncertainties of the constraints. This paper tackles the filtering problem of dynamic systems subject to the stochastic linear equality constraints expressed by random weighted basis matrices. The corresponding constrained dynamic model is constructed first and the linear-minimum-mean-square-error filter is derived based on the orthogonality principle. Due to the effect of constraint randomness, the resultant filter encounters the problem of nonlinear stochastic calculation of random parameters, which is solved by the Taylor-based and the UT-based schemes, respectively, and the computational complexity as well as the tractability of both schemes are analysed. Finally, a simulation study on a road-constrained vehicle tracking demonstrates that the proposed filter has better performance than the classical estimation projection method in terms of estimation accuracy and computational complexity.