Based on On The Spot comments, I did a search and the places where you can get fingerprinted, in addition to virtually any US sheriffs office, are listed here:
http://www.tsc-csc.com/printoffices/
There are indeed a few airports listed. Unfortunately, if you go to any of the US-based JAA/CAA-certified flight schools, either in Florida or California, it doesn't seem that the obvious airports for getting there have TSA offices.
On The Spot went to the US for a type rating, not a PPL, as I understand. It may well be that there was a TSA office conveniently located where he did his type rating. What is important to remember though, is that a type rating may well mean that you're not a "Category 3 Alien" but a "Category 1 Alien", which means that the total background check has to be completed before you're allowed to start training - this may take up to 30 days, I think, after the TSA has received your prints.