The 74HC595 shift register has a maximum sink current of aprox 25mA which is too small. Another choice might be the HEF4794 which is know as an 8-stage shift-and-store register LED driver, however, this also seems to have a relatively low maximum sink current.
I found reference 1) to the MIC2981 which is a high-voltage high-current source driver array. I think this can be used to drive the coils with the shift register as the input (possibly using the 74HC595).
The clear winner looks to be the TPIC6B595. This is a high-power medium-current shift register which has a continuous sink of 150mA with a 500mA limit!!! Since each coil only needs a short pulse of current, this chip might be ideal. The absolute minimum current required to flip a disc needs to be determined.
Found reference 2) in a document describing a land mine that uses flip disc displays. The document claims the coils require a 250mA pulse for at least 1.5ms.
In my testing, 250mA from a 5v source works.
Possibly a gate, flipflop, transistor or mosfet
With help from liggyman on #hacklabto this looks to be a transistor in a SOT-323 package (NPN?).