How can I repair/rehab this old veneer door?

You could try adding some material to it like we did with a door in our place. It could hide the pealing laminate and give some personality.

https://www.frugalfamilytimes.com/2019/05/our-easy-hollow-core-door-makeover.html

Hi Barb. Well, it may be a hollow core door. If you strip the veneer, you may have nothing left. If it's "solid" mdf you may have a crumbling mess since old mdf deteriorates over time. How about sanding off the high spots and resurface with a thin coat of Bondo. It's commonly used on car body work and sand-able. Then repaint with an oil base paint. Will last a lot longer in humid climates. Good luck.

My door in my last house was solid wood, but was pretty old. I installed a brass kick plate at the bottom of it to cover the lifting veneer. Then I painted just the window frame black and opened the lifted veneer and injected wood glue into the cracks that were most obvious. I clamped a board across the door for the afternoon to keep the glue in place. Then I lightly sanded and re-sealed the wood. I kind of think that your door might be hollow, however, which wouldn't do as well as mine did. It looks as though your window is the culprit for moisture going behind the veneer. Whatever you do, be sure to completely clean and re-seal that window with silicone before performing any treatments...

I believe that person has aged out!!!

I lifted the veneer back, there is wood underneath but is hollow core door. The there definitely is wood under the veneer. Should I remove the window caulk and reinstall?

Barbara, You've got a number of things to consider here:

1. If you're on an island with lots of wind, unless you're willing to seal the jalousie windows permanently closed, you'll likely always have water problems.
2. Given the amount of damage to the veneer, my guess is there's hidden wood rot inside the door which will require more work (see my articles on wood rot repairs).
3. You seem to know you've got a hollow door & they're much more difficult to repair because they're made from plywood or MDF, which is very thin.
4. You really should spend your time finding a replacement door as refinishing doors is a lot more work than your realize (an article on steps just to remove veneer).
5. You might be confusing doors in inventory on the island vs what can be ordered. I would recommend finding a fiberglass door to withstand the weather & order it from a store on island or online ... but you'll want to talk to folks at the store for who can install because doors are tricky (tips on ordering an exterior door).