...Hard to imagine a "tool chest" with casters.
Tomwil: I have a hard time believing that a guy posting on GarageJournal.com can't imagine a wheeled “tool chest”. I hope you are not going to devolve this into a discussion of semantics about the differences in a tool box, tool chest, tool cart, etc.
To the OP I have a couple of ideas:
(1) While it is not exactly a drawer, have you considered an appropriately sized sheet of plywood, sheet metal or baking sheet with a few ball rollers (lower profile than wheels) screwed to it.
That would allow you to place flatish things that won’t fall off when you roll it under. Think of it as a place to store some of those tools that have come in blow mold cases.
(2) Screw a piece of aluminum channel to the bottom of the box going front to rear just inside the caster with the "C" facing inboard. Place a second piece towards the center of the box with "C" facing outboard. Space them just wide enough to capture the lip of a plastic restaurant dish tote. The channel will effectively act as a runner for the tote to slide in an out.
The totes come in many sizes and are relatively inexpensive. I have found the better quality ones come from restaurant supply houses. They are great for storing projects that are "in the works".