I have the ATI All in Wonder 9700 Pro. The TV card has the ability to
show closed caption either on the TV picture and/or on a separate
window. What I would like to know is whether I can save the closed
caption to a file. And if so, can this be done while simply watching
TV or does the show need to be taped. As an end result, I would like
to see a text file of the closed caption transcript.

5 stars if you can demonstrate that this is not possible.

5 stars and a 100% tip if you can show how this can be done.

Yes that can be done

It is called the TV Magazine function

It the last bersion of Multimedia center I used all it could do was
save the caption text to a txt file but I understand that nowadays you
can even add still shots

To enable it go to the tuner applet select the close caption tab and
mark the box tv magazine on the top right panel
then select the button underneath to set up what you want to capture

Thank you for this excellent comment, corwin02-ga. It works precisely
as you described.