Must a thumbprint accompany a notarized document? 

To deter fraud in notarizing, thumbprints are sometimes required by law in certain states, but are always a good idea.  California notary law stipulates that the notary must take a journal thumbprint when notarizing signatures on powers of attorney or deeds effecting real property such as Grant Deeds, Quit Claim Deeds, Mortgages, Subordination Agreements, etc.   Other states have their own rules.  Texas has some rules restricting the use of thumbprints, but I don’t know enough about those restrictions to comment.
 
Prevent fraud
As a general rule, if the notary public you use takes a journal thumbprint (many do not bother with this or even own a thumbprinting pad), you have more security.  The thumbprint is proof that nobody faked an ID and pretended to be you, or forged your signature.
 
Serious documents should have a thumbprint
If you are having a serious document notarized, you might ask ahead of time if the notary carries a thumbprinting pad.  They are two inches in diameter and weigh about half an ounce, so it is not a burden to the notary, assuming he/she is prudent about notarizing (that is assuming a lot).
 
Does the thumbprint go on the actual document?
I have never heard of a procedure which requires a thumbprint on an actual document, but it is not a bad idea. You could neatly put it to the right of a signature and document which thumb was used from which individual.  If you are missing a thumb, you can use the other thumb or a finger, just document it somewhere.

You might also like:

Notary Public 101 – identification
http://blog.123notary.com/?p=19507

Notice to title companies about thumbprinting
http://blog.123notary.com/?p=19453

Identification and thumbprint requirements for notarizations
http://blog.123notary.com/?p=4299

Signing agent best practices
http://blog.123notary.com/?p=4315

Notary thumbprints can save your neck
http://www.123notary.com/forum/topic.asp?TOPIC_ID=4939