Sometimes when I am collecting resources, an animation comes up on the bottom right of a dweller carrying a bunch of boxes looking stressed. What does that mean? Am I doing something wrong?

By memory I've gotten that animation when I am getting close to capacity in my storerooms.

Exactly that. It means your storerooms are getting near their max capacity,

I have gotten it with even 100 storage spaces free. So I'm not sure if that's why it pops up.

Yeah that was my assumption Yorien

That's your stimpack and radaway storage that's gotten full.