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!

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 Side 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!

Here is a list of all characters with moves that will generate an item that other players can grab. As I don't have a switch with me, I won't be able to test some of them to make sure that the staling properties work as expected. I'll be updating this answer in a few hours when I'm back home.

Bowser Jr.

• Koopa - Down Special

Diddy Kong

• Banana - Down Special

King K. Rool

• Crown? - Side Special (Maybe only when it's been grounded? This needs to be tested...)

Link/Toon Link/Young Link

• Bomb - Down Special

Mega Man

• Metal Blade - Neutral Special

Pacman

• Bonus Fruit? - Neutral Special (Needs to be tested for staling properties)

Peach/Daisy

• Turnip - Down Special

Rob

• Gyro - Down Special

Robin (Both of these depend on OP's definition of "special move counting as an item)

• Tomes - All special moves
• Levin Sword - From overusing aerials

Simon/Richter

• Holy Fire/Aura - Down Special

Snake

• Grenades - Neutral Special (Needs to be tested for staling properties)

Wario

• Bike - Side Special (Needs to be tested for staling properties)

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!