I think Sethbling did a concept like this before, but here's my attempt:
Much like my Shulker box bank, you open accounts with copies of a written book that way only you can access your account. You put your "keycard" book into the dropper and press the button. The minecart shows up on the rail to the left and you need to wait for it to go back, which is indicated by the lamp on the side.
Those yellow circuits are item sorters for the "keycards," the hopper lines simply transport items, the lime things are just structures to hold the minecarts, the magenta is the minecart line to the player, the grey is the minecart line back to its holding position, the light blue is to detect if the minecart is has left its holding position, the orange releases the minecart, the green if for the lamp indicator, and the blue is to seal off a holding position if it already has a minecart in it.
Here's a detailed look at one module:
This design could be compacted a bit more, but I think the proof-of-concept is pretty cool.