yes you are right about the 1/2" being need for support on my chairs, but a 1/8" staple will not even penetrate a folded piece of your fabric
1/8" thick would probably have to be purchased at a hobby shop (not cheap) and not in the size you need I've never seen less than 1/4" at lumber
yards or HD, or Lowes. Besides when you sit on the top it pulls on the sides and 1/8" would curl like wet lasagna.
I am just offering an opinion but I have spent 42 years in many phases of home building, rebuilt an entire 35' Chris Craft including made new decking with caulking grooves,rebuilt 2 engines and installed them. Restored 3 antique autos and upholstered them. I do not want this to sound like an I did I I I rant, just want to be clear I do know a little about this and I hate to see people waste time and money
The staple must not only go through the fabric
it must go through (come out the other side)of the wood on anything under 3/8".
You might be able to get away with 1/4", but 3/8" would be better, but I will use 1/2"birch ply the US made stuff has exterior glued plies. Do not use cheap
(that wavy, curled stuff you see at HD) or you will have wavy bases.