Quest rewards no longer fall at your feet when your satchel is full. If there isn't room, the quest-giver asks you to make room first, and you can hand the quest in once you have. The same goes for quests and characters that hand you items: they wait until you have room.
Lost your Moorland Pony's halter? Odile Fairweather at the Reedlight Inn keeps a spare for anyone who brought Bramble home.