Does anyone know of a hard, heavy, moldable material (like plaster or cement) that is also food-safe? I’m looking for something to make bowls and cups without kiln firing like I would have to with clay.
Food safe, moldable materials are rare in general. The closest thing that comes to my mind is a kind of "organic plastic" made from milk, but it looked hard to work with and I don't know how food safe that really is on the long run.
Take for example a simple porcelain dish: the porcelain itself is porous and would stain with food residue. It's the glaze that makes it food safe.
I suggest using whatever clay is best suited for the style of bowls you want to create, then applying a food safe varnish or lacquer in several coats to seal the surface.
"Hard, heavy, and moldable like plaster or cement" sounds like the intent is to cast the items rather than shape them by hand like clay (although there is cement that is moldable like clay, like Sakrete's ShapeCrete). There are some materials that cure at room temperature and would be strong enough to use for bowls or cups (although more fragile than clay or porcelain).
For example, CementAll from Rapid Set is a type of fine cement often used for casting art objects. Jesmonite is a high-grade plaster mixed with an acrylic resin, which is much tougher than plaster. It's used in construction as well as casting art and utility objects that will get handled. There are a few other options.
Making it food safe
The problem, as Elmy's answer describes, is that none of these are food safe. You can seal a concrete countertop to make it "food safe", but it isn't the same kind of food safe as cups and bowls need to be when they will be holding food or hot liquids and you will be eating and drinking from them.
Also, there is no kind of air-dry sealer or coating that will be equivalent to a kiln-fired surface in terms of longevity. Elmy's answer talks about using a food-safe varnish or lacquer. You could probably give a cup or bowl a food safe surface that way. The problem is that while those finishes might, themselves, be food safe, they aren't designed to make something permanently food safe in active use. The cups and bowls wouldn't remain food safe for long.
The kiln-fired surfaces basically turn into a kind of glass, which is hard and tough. If you notice grey stains in the bottom of your coffee or tea cup, those are deposits of metal from your spoon because the surface of the cup is harder than the silverware. If you don't break or chip the cup or bowl, the surface will stand up to decades of use, utensils, and cleaning, and will still be food safe.
There is no air dry sealer or finish you can apply that will come close. What is used to seal concrete countertops soaks into the concrete pores to seal it against liquid penetration, and leaves a film of plastic on the surface. That makes it cleanable. If the surface is clean and you drop some food on it, the food will still be safe to eat. But you can't use the surface to cut food on, that would scratch the plastic film. And cleaning the countertop and sliding stuff around on it abrades the surface film, so you periodically need to seal it again.
If you make cups and bowls out of these kinds of air dry materials and air dry sealers/finishes, you will have fragile cups and bowls that periodically need to be resealed to remain food safe.
A possible solution
An alternative that is strong and durable, with room temperature curing, would be to cast them with a food safe resin-based material. If you use just casting resin, the result would look like plastic rather than porcelain or clay, and it would be much softer. It would get scratched and abraded with use.
However, you can mix in inert mineral material, like sand. Make it mostly mineral material, with just enough resin to act as a binder. The result will look very different, closer to mineral than to resin. It will also be hard and tough, more like rock than plastic. Corian countertops are a somewhat similar material.
One thing to be aware of with these resins is that being food safe requires perfect curing. Implementing this to produce a food-safe end result would be subject material for a different question.