An internet search has lead me to this "history" lesson.  It appears that the term found its start on a college campus or within the political arena.  Either way it appears simply a cheap trick when all else FAILS.

This came from a search on MAMMA.com a metafile search engine.

Posted by Paul Guthrie on September 04, 2000 at 14:58:33:

In Reply to: Re: play the race card posted by ESC on September 04, 2000 at 04:16:06:

: : : :
: : : : "Play the race card" has been variously identified as a recent invention and as a coinage of the 1960s. Does anyone have a citation earlier than 1996?

: : A couple more 'playing the race card' references both dated 1991, the earliest I have found, but still nothing from 60's (or the 70's or 80's.)

: : 1991 - Michigan State University
: : 'Community Policing and the Challenge of Diversity'
: : Both Louisiana's David Duke and New York's Rev. Al Sharpton are skilled at playing the politics of racial resentment. Racial stereotyping allows those who play the "race card" to manipulate audiences by tapping the "hot button" issues based on bias.
: : http://www.ssc.msu.edu/~cj/cp/diverse.html

: : 1991 - Time Editor-at-Large and former Washington Bureau Chief Strobe Talbott on Inside Washington
: : "I think the argument for rejecting him has more to do with the President who nominated him than Judge Thomas himself. This was an entirely cynical nomination to begin with. It was an attempt to basically play the race card by picking somebody who, because of the color of his skin, would not be rejected, even though he stood for a set of views on society and how the justice system would work that are completely against those of the Justice he's replacing, Thurgood Marshall."
: : http://www.mediaresearch.org/news/nq/1991-1992/nq19910916.html

: My 9/1/00 post has a citation from the Carter administration.

Just thought you may find this information interesting.