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Two phyla of animals use a gastrovascular cavity[1] in which food enters and waste exits through the same orifice.

1. https://en.wikipedia.org/wiki/Gastrovascular_cavity

I don't know if I'm missing the joke, but why do we assume the 2D stork people have a digestive tract and not a series of manipulatable pores that facilitate the movement of particles in and out of the body or maybe similar to how an amoeba uses phagocytosis.

not necessarily. instead of a tube, it would be a series of "locks" like a canal.

Regurgitation solves that specific showstopper.

https://users.renyi.hu/~daniel/tmp/noitdoesnt.jpg

That's a pretty slick solution there.

Your two separate pieces could still interlock, so that you wouldn't "fall apart".

you assume a 2d being needs to need or requires a digestive tract. What if that being doesn't necessarily it?

Food may also serve as the glue.