Not sure but try pulling the bottom drawer out / off, and look in the cabinet. Vaguely remember someone mentioning it. If all else fails give a local sales guy a call and ask him. I'd lead off with "My first question is how do I get a catalog" they're more apt to do more for a potential customer. If all that doesn't wrok for you GOOGLE is your friend!
The early boxes did not have serial numbers. If your had one it would be stamped into the back panel with the Snapon Company info. Yours looks to be pre 1960 But not before 1956 My guess
You can go to GarageJournal.com website, there is a vintage tool forum there. You can join the the forum and post pics of your toolbox. Lots of knowledgeable people there.