No, they won't start the mission armed, but they would have a chance to pick up weapons dropped by dead/stunned/panicking soldiers/aliens and use them, a very small chance since 1) not even the (default) "fighter" AI actively seeks dropped weapons — much less the civilian AI — only if they happen to end their turn standing on top of one they'll try to pick it up the next round, but 2) civilians don't really have combat capable stats, they might not have enough strength to pick the weapon up, or not enough TU's to use it (specially if they get an encumbrance penalty) even if they can hold it.