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.
(To vote for me, please hit the (+) plus button near the top of the page on the right!)