Go to a real lumberyard. They should be able to help (stay away from HD, Lowes, etc) Ask about Johnson Hardware and the thinnest door you will get in good quality will be 1-3/8" thick.

I really just need a flat panel more than a "door" door--would you advise against anything less than 1 3/8" for any reason? I just don't think I have the closet depth for that; it's quite tight.

You need to use a door if you want it to remain flat, especially as a slider as it will only be fixed at the top and basically hang. They do make 1-1/8" doors (I believe they are still made), however they were generally the cheapest crap available. So in answer to your question, no I would advise against anything other than a 1-3/8" door at the thinnest!

Hi, I have seen hardboard panels with a metal frame they went from ceiling to floor. They looked terrible. If space is a problem put up a travis rods and hang some drapes on the wall outside the closet. You won't lose any closet space
Woodbutcher