I don't know how to disassemble the axle to get access to the bearings. If there's no clear flats, or a 6 or 8 sided nut visible, then you might be looking at a one-time assembly.
Try turning the rosette-end of the shaft visible in the photos, using needle nosed pliers. I suspect it won't do anything.
However as a half-way option, try dropping some solvent in the open end, shake it around to clean out the old lubricant and dump it out to flush out sediments/grit. Repeat.
Once it has dried, add fresh lubricant, and spin the pedal for a while to help it settle.
Then find some way to seal the end of the pedal - perhaps salvage some kind of end cap from elsewhere, or even just sticking some duct tape over the end to keep the world out.
Your other option is to buy replacement pedals. They look like vintage road cleats - downside of modern ones is might not look "right".