I think this swapbox could be connected to a more modern machine, but it'd require some soldering work to achieve.
From looking at the connectors on the cable and card reader, it appears to use a pair of 50-pin IDC ribbon cables.
PCMCIA / Cardbus cards have 68 connectors each, however. There doesn't appear to be any duplication of pins (e.g. multiple ground lines), so the manufacturer has managed to squeeze 2x68 wires for the two slots into 2x50 connectors somehow. They might have done this by not duplicating some wires used by both slots (e.g. the 25 address lines).
At the "modern PC" end, PCI adapters for PCMCIA / Cardbus cards were (and are still) commercially available. They tend to be one-card adapters, rather than two-card ones, so you'd need two of them if you wanted to use both of your swapbox's slots.
That leaves the trickier issue of connecting the swapbox to the new adapters. This would probably require making a new, bespoke cable. The swapbox clearly uses a non-standard pinout, so you'd need to reverse-engineer this part first.
Does the swapbox have any chips or active components inside it? It'd be worth opening the cover to have a look first. If the swapbox is just a passive adapter, it would be possible to use continuity test function of a multimeter to work out how each slot's pins are connected to the IDC connectors. It'd be a slow job, but not a complicated one.
Once you know the pinout of the swapbox's IDC connectors, you'd need to come up with a way of connecting it to the PCI adapter. Bare PCMCIA connectors can be bought new, but they're relatively expensive for what they are. I'd suggest using the PCI card as-is, and cannibalising an inexpensive CompactFlash / Microdrive adapter for its PCMCIA connector instead.
This could be soldered to a ribbon cable leading to the swapbox, using the pinout that you created earlier.
If, on the other hand, the swapbox has some active computer chips inside, connecting a PCI-PCMCIA adapter to the IDC connectors may not work correctly: the PCI-PCMCIA adapter would expect to be connected to a PCMCIA card, not to an intermediate interface chip.
If you're able to remove the swapbox's case and photograph the circuit board inside, it may be possible to give a more definitive answer.