First, a Wiki is a simple online database that's easy for multiple people to add to, and edit, from around the world. Think of the free online encyclopedia Wikipedia
Now, to answer the original question, you don't need one...you can actually use your own computer to get the job done. Just click here for the steps, and the free software you'll need. If you would still like to use a web host, just do a general web search for ones that start as low as $5/month. (I don't have any recommendations, though someone else might.)
Hope this helps,
John