I have always found that some L Shape parts that you can clamp in the vise are very useful. For this particular situation, you could clamp an L piece in your vise. Screw together a step piece from some scrap, and put it in front of the bench. Grab another piece of scrap to use as a brace for the far end. Like in the picture attached. If you box clever with the step piece dimensions, you can build just one, and rotate it when you want to switch from planing the long sides to planing the short sides. You can double stick some old rag, or cork or anything else to the l piece, brace and step to keep them from damaging your drawers. If you want to have some support in the middle of the drawer (not quite sure why you would if you are only working on the joints) take two pieces of wood, both of which would be smaller than the gap you want to support, and screw them together to match the gap you want to support.

That method works even if the vise isn’t flush. Not a problem. The drawer is just held between the two jaws and the clamped piece acts just as in the drawing. An alternative is to use a bucket, stool, or other improvised jack instead of the wooden clamped pieces. This is helpful when there is a bottom and in some other cases.

“Just put the open side of the drawer facing the workbench.”

That definitely works. If the grain is against you, you won’t be able to flip around, so just pull the plane instead or develop your feel for planing with your off hand. I’m really, really glad I’ve been developing the ability to plane with either hand! I’m a lefty. Some tools, especially old molding planes, have made me develop my right hand. It feels odd at first, but it is so very much worth it. I don’t think I’ll ever be able to swing a hammer with my right hand, but a lot of other stuff works just fine. Anyway, if there is a bottom and you do and you use Mr. Van Noten’s excellent advise, just be ready to come at your far side with a pull, your off hand, or cross hip.

To Edmund
I once was standing on a deck while the the top of a door to be planed was resting on the ground a few feet below (to plane the bottom of the door).

You might have noted that Paul Sellers does not necessarily works vertically or horizontally.
See for example how he planes the ends of the work-top and aprons of the workbench. (see introducing picture of the video “workbench: episode 1” [the solid wood one])

You could also have been standing/kneeling down on your bench.

See 2nd picture in “will you apprentice with me?” dated 4 August 2015; although I would not have done this for that box.
And John’s piece doesn’t seem to be clamped except maybe with his foot and the piece against the wall. (it probably works better with a very sharp plane).