When did The Daily Show (with Craig Kilborn) episode air when they did
a story on the Osmonds' Christmas, featuring a "funky Santa"?  (My
guess is that it was around late 1997.)  Also, how can old episodes of
The Daily Show be legally acquired?

Well, Donny Osmond was the special guest for the "Holiday Spectacular"
on December 19, 1996 and returned the next year for "The Daily Show
Andy Williams Christmas Special" which aired on December 17, 1997. 
Craig Kilborn was on suspension for the latter show and A. Whitney
Brown filled in for him.  I don't remember the segment you mention
specifically, but it seems to me that a story on the Osmonds'
Christmas would have more likely to air on that special than any other
episode.

Unfortunately, other than the Indecision 2004 DVD, I'm pretty sure
it's impossible to obtain old Daily Shows legally.