Well, the original patent by the guy that invented the Magnatone Vibrato shows a bridge circuit where the arms are a cap and a "variable impedance element". When that variable impedance changes and the cap value stays the same, the phase angle for a given frequency changes. A changing phase mixed with a constant-phase (dry) signal gives a varying pitch, akin to Doppler effect.
As the Magnatone incorporates the invention, that variable impedance is a varistor, which alters its apparent resistance in response to a changing applied voltage. That "changing applied voltage" is the signal created by the oscillator, just like a trem circuit. The difference is the application of the oscillator signal varies signal amplitude in a trem, while it changes phase, and thereby pitch, in a vibrato circuit.
It's worth noting the Univibe uses LDR optoisolators as its variable impedance element. I've been thinking we should be able to use Vactrol optos as the variable impedance (there's a whole range of these devices; Doug sells at least one type).
The catch is the control signal has to be applied slightly differently, because it won't be applied directly to the controlled element, as in the Maggie vibrato. Also, the size of the control signal may need tweaking and/or cap values in the bridge circuit may need to be adjusted.
It's something that's worth experimenting with, but it needs breadboarding to zero-in on good parts values and overall circuit implementation.
I've just seen the original varistors selling for something like $75/pair, and you need 4-8 of them depending on which specific Magnatone model you copy. To me, that seems unconscionable, and there's got to be a better way...