When your mayor or other characters have a birthday, the villagers will celebrate it with you and give you one item from the set. You can only celebrate once per year even if you tt (I would guess if you tt in and out of the year you could reset that but no idea).
So if you want to get it all by yourself, you can get up to 4 random items a year. If you don't need all your alts you can probably get one every month or two (you have to be pretty friendly with at least three villagers) -- I would guess that's how people put the set together quickly.