To answer your question, I think it's good to understand the meanings of "窝" mentioned here.
Three possible meanings of “窝" used in this case:
to refer to several animal's babies who are born at the same time.
This could happen to the mouse,pig, rabit，chicken ,etc. For these animals, the mother animal usually can give birth to(or brood, as a hen does) several babies at a time; but for animals like the cattle and tiger, the mother animal usually(exceptions are rare to see) can give birth to only one baby, so it sounds very strange to say "一窝牛" or "一窝虎", but sounds good to say"一窝兔子”,"一窝小鸡".
In this case, because it's used for animials, and more importantly, parents and children always can't be born at the same time, by the same mother. You should not use 窝 to refer to a family, even in a figurative way.
to refer to a group of bad people or things.
一窝土匪 a group of bandits
Remeber, Bad people and things, so 窝 is also not acceptable in this case.
animal's burrow or nest
In this case, 窝 seems accpetable to mean "house, the place where a man lives ", and could be used to refer to a family since all the members are in the same house.but as previous answers pointed out, it may offend someone. Don't risk saying that way!
I have never spoken that way or heard of someone saying that way in the real life.
So how to describe a family when all the family members are of the same zodiac animal?
I would like to say"他们一家都属[X]" or "他们一家都是属[X]的". Here, X is the name of the animal, 鼠，牛，虎，兔，etc.