Here is a list of all characters with moves that will generate an item that other players can grab. I tested all of them in training mode. Unless noted otherwise, none of these will stale:
- Mecha-Koopa - Bowser Jr. Down Special
- Banana - Diddy Kong Down Special
Crown - King K. Rool Neutral Special- This move will stale when used as a special boomerang-style. While it's possible to pick up and throw if you don't get it back, unfortunately K. Rool can't pick up the crown himself to use as a projectile, he just places it back on his head.
- Bomb - Link/Toon Link/Young Link Down Special
- Metal Blade - Mega Man Neutral Special
- Bonus Fruit - Pacman Neutral Special
- Turnip - Peach/Daisy Down Special
- Gyro - Rob Down Special
- Tome/Levin Sword - Robin Specials/Aerials
- Depending on OP's definition of "special move counting as an item"
- Holy Fire/Aura - Simon/Ricther Down Special
- Grenade - Snake Neutral Special
- Wario Bike - Wario Neutral Special
- Throwing the bike will never stale, but riding it into people will.
That's all the items that I was able to find, but if I've missed any you think might work, let me know and I can test them out!