It's possible, but no one does it. Notice how the construction guys call it a day when it starts to rain? There's a reason for that -- you're asking for trouble if you erect a roof in the rain. Don't do it. Just wait til it clears up.
It's possible, but it's not recommended. No mater how careful you are and what temporary roofs or tarps you erect, your materials are more likely to be damp and affect the structural integrity of the building.