The subject should be 'each person' or 'everybody/everyone', and the verb should be 'take'. 'Carry' would be the wrong word here. 'Carry' would refer to the act of walking with the objects in your hand/on your back, rather than the responsibility of providing the food. I'd instinctively use the verb 'bring' here, though. As we talked about in one of your previous queries, 'bring' suggests a movement towards a focal point. Here the focal point is the picnic, so I'd use 'bring' - although 'take' is also ok.

*Things for a picnic* I would say it like this, "Whenever we go on a picnic everyone brings something. Last time I brought some cheese to share". Instead of "each one" I would say everyone brings something. If everyone carries something that means that the item is physically on their person. Which if that is what you are trying to convey I would say like this: "Everyone carries whatever they brought for the picnic". Hope that makes sense!!!