I've already got Piper on my first vault, but I am thinking about another one for my second vault.

Is it possible to get another one from lunch boxes?

I've got multiple Preston Garveys, you would think it would follow that Piper would be available.

However, if I had a lunchbox for everytime that something 'you would think' would work a certain way with this game doesn't actually work that way, I'd have enough lunchboxes to test whether Piper was actually available.

Yes. You can. Also Mr Handy is possible from a lunchbox. Every item/dweller is.