Yes -- you can piece it together and (eventually) get the whole set. But it won't be cheap.If you’re really hard set on obtaining, why not try buying a few sets on ebay and hopefully together you can make one complete set?
I wanted the Fisher Price Noah's Ark for my grandson (not nearly as hard as your goal), and I ended up buying from about four different buyers. Now he has the ark + 18 pairs of animals. I got a few duplicates in the process, and I have them bagged up /intend to give them to Goodwill.