Dense, hard bread was used as plates, historically - trenchers - even for foods that can be generously sauced. Some of the liquid might soak in, but the structure should remain sound.
I would imagine a dense cracker, like hardtack, could be shaped into a cup and baked hard, and then used without leaking. A fluffy/airy cracker would sog up pretty quickly, and in a richer dough fats might lift out with heat (and leave openings for liquid), but a simple flour-and-water dough could probably be made sturdy enough to hold liquid without dissolving for some time. Maybe like salt dough ornaments, only without enough salt to be inedible? Perhaps sugar instead? or edible play-dough recipes, may become sturdy after baking? Should work for "sturdy", anyway.
It would be tricky to balance sturdiness with palatability, though, as a generously sturdy cup would tend to be both tough and tasteless (and dry), a thinner one my be less sturdy, and one that can soften or gain flavorings from the liquid would have to be carefully timed not to soak too quickly.
You might manage something with an unleavened sweet cracker, or a very hard/crunchy cookie, rolled out on the thinner side so it can be comfortably bitten into once the liquid is drunk. Perhaps spices can give it flavor, as too much sugar will disrupt the cracker's consistency as it dissolves. It may be helpful to dry before baking, as is done with clay, to prevent cracking or the formation of air pockets... since docking will be counterproductive. Or perhaps baked twice, to draw off extra moisture.