How do I sand an old walnut tray?

If your tray has sticky spots, use a degreaser like goo gone or something similar and wash and rinse the tray. Let the tray dry, then you can sand wet or dry depending on the type of sandpaper. If you are going to paint it, be sure and use a good primer first.

Yes to what debi said! You can also use a small multi tool that has a sanding tool attachment for any hard to reach spots.

I would use dry sandpaper.

For wood, use dry sand paper. Wet sand paper can fray the wood fibers. Wet sand paper is meant for metal or glass.