Well, to make caviar (and you haven't read the book Jewels of the Sea), you need to make a fish pond and put a sturgeon in it. It will ask you for items, and if you give them what they want, their numbers will grow and make them more likely to give you sturgeon roe, the only roe that will turn into caviar when put in a preserve jar. Any other roe just turn into aged (type) roe. If you've got golden animal crackers you could give one to the pond to make sure they give you 2 roe when they finally do, so you can get a caviar for the shipping list too.
If you have read Jewels of the Sea, you could try to catch sturgeons in summer or winter until you manage to get a chest with roe in it together with the fish.