In this day and age the Internet is a great place to start. The obvious thing is to search sites like eBay, Craigslist, even Amazon and so on. But you can make those sites work for you as well.
For example post a desription of the old and rare stuffed animals you a looking for in the eBay Want It Now section. If you have pictures, even better. Note the price range you are willing to negotiate. This will make it easier for sellers to offer what you want.
You can also add the toys to the eBay favorite items list. This way ebay will look for such listings and notify you when one pops up.
Don’t neglect the power of search engines. Use them to find more information about the toys and possible listings. Sometimes people sell such items on special forums.
Still no luck? Visit thrift stores and garage sales. You never know what you may find. Including those old and rare stuffed animals you thought were lost in time. Some specialized toy stores even help track down specific items and delivering them to you. Well, it may cost extra, but negotiate.
Another way to find old and rare stuffed animals is to check antique stores. They also can help you find the items you are looking for. This way might be the most expensive one, but usually yields results.