As I was thinking on this I eventually settled on an accordion envelope with thin material for the accordions and constrained at the bottom to allow it to flex open and display the contents all at once---random access. This is a common design, and if the accordion material is thin enough, it stays pretty thin overall.
The materials used here (paper) were selected for ease of manipulation, but something with more strength or resistance to moisture could easily be used, either for the cover, the inside, or both. Fabric, metal, even panoramic photos could be used for the covers. The interior is best kept thin, and the paper I used could easily gain some moisture resistance from scotchguard. Closing the wallet is also wide open to change. Two closures are shown, but many are possible, especially with more robust cover material.
Final thickness of the empty card wallets are about 0.5 cm and can hold from 10-13 cards.
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Step 1: Gather Materials
Glue (I chose mucilage for its paper-bonding as well as its applicator)
Foldable material for inner accordions. I chose thin graph paper. .25" or .20" squares
Stiffer material for cover. I chose an old brown mailing envelope and a 20lb bond sheet of paper.
Ruler: To measure and as a fold-starter.