The answer is no. The alt would need to have the same amount of fame required to use the purchased pieces.
What my learned friend means is YES they can use it by putting it in archive but you would still need the required fame amount to equip it
Toon 1 gets Dagger requiring 2000 Fame to "buy" it
Toon 2 would need 2000 Fame with the same faction to equip/use it