Sounds like the locking mechanism on the pedestal door is not holding and the slowing down of the spin (where it shakes) is causing it to pop open.

"It appears level because when I try to shake it, it doesn't rock."

That doesn't tell you whether the washer is level. If you don't own one, go to Home Depot or a hardware store and buy yourself a level. Check that the washer is level front-to-back and side-to-side. Adjust the washer's feet to level the appliance.

There is no locking mechanism on the drawer. It’s just pushed shut and it glides on rollers, so it doesn’t take much vibration for it to start rolling open on its own.

The washer is level and I'm still finding the drawer open like this after a load of laundry completes.

How about buying a touch latch and attaching the latch and plate to the inside top and inside front of the box, respectively with shallow sheet metal screws?

Do you have anything stored in the drawer?

Try adjusting the washer to be slightly "un"level, leaning slightly back, if you're willing to do that.

Or, maybe a simple solution is place a strip of tape on the drawer to hold it shut.

Maybe try some velcro with one piece stuck to the top back (inside) of the drawer's face piece and the matching piece on the corresponding point of the front lip of the drawer box?