These guys aren’t cheap, but they have 1/16 birch:
https://northeasternscalelumber.com/shop/plywood.html ($7.09/sqft)
Here’s a thread on another forum about it:
http://lumberjocks.com/topics/80249
From which we see that national balsa has 1/16" for about half the price of northeastern scale lumber: http://www.nationalbalsa.com/birch_plywood_s/138.htm ($3.52 /sqft)
Then I found this - http://www.boulterplywood.com/MarinePlywood_4.htm Their price for okume ply varies based on size. 4x8’ sheets go for cheapest ($1.38/sqft) and 2x4’ sheets are a bit more ($2.25/sqft) You’ll probably get absolutely housed on shipping though.
http://www.aircraftspruce.com/catalog/wppages/finnishbirch.php have various cuts too, the half sheets are 4x4’ ($1.20/sqft) and full sheets are pretty much identical price per square foot.
http://shop.balsausa.com/product_p/346.htm has 1/16" ($3.94/sqft)
https://www.towerhobbies.com/cgi-bin/wti0095p?FVSEARCH=plywood+1%2F16 are expensive, about $10/sqft.
https://cherrytreetoys.com/1-16-birch-plywood/ ($5/sqft)
https://sigmfg.com/collections/plywood/products/sigpw005-plywood-1-16-x-6-x-12-3-ply?variant=12308916600910 ($5.33/sqft)
http://www.balsatron.com/Item/1-16-x-12-x-24-3-ply-birch-plywood ($6/sqft)
https://www.horizonhobby.com/product/airplanes/tools--equipment--and-supplies/wood/plywood-1-16-x-12-x-48-(6)-mid5482 ($6.88/sqft)
I called these guys: https://forestplywood.com/product/baltic-birch/#specifications and got a quote of $58.50/sheet ($2.34/sqft) but they don’t ship. If you’re near San Diego, though, that’s a good price considering there wouldn’t be shipping involved. They have 1/8" for $11.40 ($0.46/sqft !! ) a sheet too, which is the new winner for 1/8" price.
https://www.greathobbies.com/productinfo/?prod_id=P2411612 seems Canadian, ($8CAD -$6.09USD/sqft)
http://www.aircraftplywood.com/products.html These guys are a manufacturer, so when i called he chuckled at my question about the order size. But, if you want to go big, there’s a 25 sheet minimum at $65/sheet ($2.03/sqft, but total order of $1,625.00… AND shipping, ouch)
http://www.hyatts.com/art/plywood-1-6mm-1-16-x-12-in-x-24-in-X77531 ($8.something/sqft)
And that’s where I ran out of steam. National Balsa, Aircraftspruce, Boulter and Forest Plywood are definitely the cheapest per square foot. I think that the bottom line will be that shipping is going to kill you no matter how do it online. If you can find a local distributor who is willing to stock it, that’s probably going to win handily.
Quick Update:
Aircraft spruce has great prices and the shipping isn’t awful if you do a couple sheets. 1 sheet is $20, but 2 is like $23. What was particularly interesting to me is that they listed an “East” location, in GA. I am in Atlanta, turns out they are in Peachtree City which is a reasonable drive for me. A field trip is in order soon.