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.
Step 2: Make the Accordions
Fan-fold pieces along graph lines, every two lines---this is the handy byproduct of using graph paper: built-in guides. This will give you a 0.5 inch deep fanfold (for 0.25 inch graph paper) or a 0.4 inch deep fanfold (for 0.2 inch graph paper). That's probably as shallow as you want to make the folds. Out of an 11 inch piece of graph paper, you get 10 "compartments" using 0.25 inch paper, and 13 "compartments" using 0.2 inch paper, with good retention of the cards.
Step 3: Make Cover and Glue Accordions on One Side
Cut a piece of cover material as wide as the widest ("tallest"? - check the picture) card you will store. For inside gluing, make it, say, another 1/8" wider. Cut the piece about 3 times the length of a business card (front, back, top flap).
Previously, when fanfolding either the 0.25" or the 0.20" quadrille graph paper (of length 11"), you probably ended up with an accordion with one extra flap, kind of like this: /\/\/\/\/\/\/\/, like an "N" with too many middle bits. Cut one of them off so you end up with something like this: /\/\/\/\/\/\/\/\, like an "M" with too many middle bits. This is more or less optional for the inside method, mandatory for the outside method.
Inside: Glue the outside of the last flap of an accordion and align it with the side of the cover at one end. Repeat for other accordion on the other side of the strip, width-wise. If you need to, Place a card to be stored into the compartment closet to the cover material to check for fit. Make sure the accordions are "mirrored", that is, that they create symmetric "compartments" between them.
Outside: Glue the inside of the first flap of an accordion and place it over the cover material at one end. The material itself and the fold help to get perfect alignment. Repeat for the other accordion on the other side of the cover, width-wise.
Step 4: Finish Gluing Accordions
Inside Gluing: Apply glue to last flap on one side, align with cover and press to apply, repeat with other accordion.
Outside Gluing: Pull last flaps up and over the cover material. Apply glue to each and press in place.
Step 5: Create a Closure
For my paper samples, I chose to make a tab for one and a captured rubber band for the other.
Captured Rubber Band: Find a light-strength rubber band that is barely snug around the width of the closed wallet. Cut the cover-flap from the end into thirds width-wise---cut maybe 1/3 of the length of the cover-flap (I cut up to 1/2, but I think 1/3 would have been better). Fold middle tab in half, glue it to itself. Fold again in half, place rubber band in the fold and glue down, capturing the rubber band. Fold side tabs and glue to give it a little extra strength.
Tab: Cut a slit into the middle of the "front", the side covered by the top cover-flap. In the cover-flap, cut a tab as wide as the slit and all the way up to the level of the slit underneath. Fold over and glue as you wish---I made a double-thickness tab with the sides folded over for some added strength.
Step 6: Finished!
Optional: While the wallet keeps cards pretty well in the friction at the base of the accordion, you can, if you wish, add some side-to-side stability by gluing down a extra piece of card-sized paper to one of the "compartments" in the middle of the accordion. You don't lose any storage space and you keep the sides from moving away from each other.