When I first got my permit (1981 was a misprint, it was actually 1983) our Sheriff did a one-on-one interview with everyone who was a first-time applicant. When my wife decided to get her permit, we renewed them on the same day (better chance to remember when the expiration date was). When I said I wanted to renew my permit and get my wife one, the Chief Deputy said "if he already has one, give her one, too." (see above statement on "questionable spouses"). This was circa 1991. Since then, it is just a matter of presenting the old permit and a check for renewal.
To answer your question, as far as I know, no background check has ever been done on me or my wife (for a permit, that is). So I can't comment on whether one is run on current applicants or not. And, believe me, I'm not complaining about the $25.00 (When I was stationed in GA, $30.00, good for 4 years).