A foam box is made in two steps. Polystyrene beads are first expanded with steam, then filled into a mould and fused together with hot steam; steam chest moulding is today's standard method. Newer plants also use radio frequency instead of steam. The resulting body is about 98 percent air, the rest polystyrene.
Density is set at this stage: the more the beads are expanded, the less material remains in the same volume. The density of foamed plastics ranges from 15 to 120 g/L, the same mould can produce boxes of different densities, and most of the price difference comes from the amount of material that enters the mould rather than the mould itself. If the specification does not state density, the quotes that come in describe different products.
So ask for quotes on the same information: material name (EPS and EPP are different materials), product drawing, dimensional tolerances and how the mass or density declaration is measured. If food contact, washing or reuse is required, ask for the conformity document separately.