I'm now figuring out what voltage to use to power the shift register.

Powering it from 5 V doesn't sound promising, as the HC164's minimum input high voltage is somewhere above 3.15 V. The Teensy, however, being a 3.3 V device, will output a high level somewhere around 2.8 V.

Powering it from 3.3 V would surely work on the Teensy side, but I'm not sure if the servos would be happy about a 2.8 V input high level (or something like that). I guess I'll have to test that...