I have an older Yubikey stashed somewhere - if I can unearth it, I'll do some tests. But I suspect it could be something as simple as a wire: on one end touching the computer's ground circuit, while at the other coiled up to make a flat spiral the size of the sensor, and having that touch the sensor. The wire could probably be isolated, because the sensor is based on capacitance, so it does not require a full circuit.
A simple way to get a ground connection to the computer is via a fake USB connector. Pin #4 is ground.
http://www.hobbytronics.co.uk/usb-connector-pinout
Or perhaps the outer metal jacket of the USB connector would also get a ground connection.
With a Mac laptop, maybe it's enough to just touch the metallic case to get a ground connection.
TLDR: A wire, connected to computer's ground at one end, with a little metal pad (or flat coil) at the other. Bring the pad close enough to the sensor, and it should trigger.
Also, most implementations of security with yubikey-type devices I've seen use it as a secondary authentication mechanism and still require a password, so it wouldn't actually solve your original issue of hands-free access. :( There may be ways to configure it to be the sole means of authentication though.
Incredibly insecure vs the OTP/U2F modes, but probably better than telling people your password.
Maybe someone you know could take the yubikey device, attach a wire to it, then connect up a relay or something that could be controlled (using whatever interface you already have - I'm not that familiar with such medical devices) to activate the relay. When the switch contacts close, the other side of the relay could be connected to an earth-grounded point. That should activate the yubikey (I believe).
Note that the wire and relay and such will need to be shielded as well (to prevent stray capacitance - like somebody walking by or such - activating it). Coaxial cable like that used for oscilloscope leads would be great, and put the relay inside a metal box (and connect the shielding of the cable to the box).
Hmm - this is getting more complicated than I thought - but I think the basics are there. Here are some other links that might help as well:
http://playground.arduino.cc/Main/CapacitiveSensor (has a nice diagram/schematic of how these form a "circuit" for the microcontroller - this may or may not be what the yubikey uses)
A couple of PDF app notes from Texas Instruments on implementing capacitive sensing:
http://www.ti.com/lit/an/snoa952/snoa952.pdf
http://www.ti.com/lit/an/snoa926a/snoa926a.pdf
Good luck, and I hope this helps or leads somewhere for you!