Storage vessels for cryogenic liquids must be well insulated to minimize evaporation caused by heat ingress. This is typically accomplished using vacated, high-performance insulation concepts such as multilayer insulation (MLI) [1]. Careful design of the vessels' structural elements is also required to minimize thermal bridging. At the same time, the structural elements must be designed to withstand stresses caused by thermal contraction and (possibly dynamic) mechanical loads. Ensuring that a storage vessel satisfies these conflicting requirements simultaneously, calls for thermal and structural modelling tools that offer great accuracy. Here we present a finite element based modelling framework suitable for estimating heat ingress and thermal stresses in MLI-insulated cryogenic storage vessels. At the core of the proposed framework is a standard, quasi-transient thermoelasticity model, which predicts heat transfer and stresses within the vessel's structural elements. This model is coupled to auxiliary models for non-local radiation [2] and heat transfer through MLI [3]. This coupling, which we achieve using Picard iterations and a single mesh for all three models, is the primary novelty of the present work. We present empirical convergence results for the coupled solvers using the method of manufactured solutions, and we study the solvers' scalability. Additionally, the modelling framework is applied to industrially relevant designs of storage tanks for liquid hydrogen carrier ships. The framework is also applicable to structural modelling of other systems where radiative heat transfer through non-participating media is of importance. [1] Ratnakar, R.R., Sun, Z., and Balakotaiah, V. Effective thermal conductivity of insulation materials for cryogenic LH2 storage tanks: A review. Int. J. Hydrogen Energy, Vol. 48(21), pp. 7770--7793, 2023. [2] Howell, J.R., Mengüc, M.P., Daun, K., and Siegel, R. (2020). Thermal Radiation Heat Transfer (7th ed.). CRC Press. https://doi.org/10.1201/9780429327308 [3] Hastings, L.J., Hedayat, A., and Brown, T.M. (2004). Analytical Modeling and Test Correlation of Variable Density Multilayer Insulation for Cryogenic Storage (No. M-1109).