Why do vibration sensors use a negative voltage?
There is no inherent advantage using a positive or negative power supply for a sensor or instrument (at least in general, it might be for a particular device in a particular application).

The reason for most (or many) eddy current sensors to have a -24 VDC power supply is, in one word: legacy.

When this technology was developed (the late 50s):


  Although the idea of an eddy-current sensor was not original to Bently (In 1958 an eddy-current sensor had already been invented by Pepperl+Fuchs as a replacement for a mechanical switch.), Don Bently innovated this technology and was the first to transistorize the design and make it commercially practical. Initially, Bently sold his eddy-current sensing systems via mail order and manufactured them one at a time. (source).


Solid state electronics was just taking off, transistors were made of germanium and it was easier to manufacture PNP transistors than NPN:


  ... Most of the earlier transistors are PNP with N type semiconductor “pellets” or “dots” of typically indium alloyed to a P type germanium wafer. This process favored PNP production as the indium had a lower melting point than the N-type germanium bases. (source)


See also this discussion on vintage radio circuits.

By the 70s, the tables were turning towards favoring NPN transistors but the need to keep components backward compatible (need that was not there for other lower-cost items like radios, tv sets or home appliances) made the negative supply the standard for some fields.

As an example, I found this circuit on a patent from 1968 showing the negative supply:



And this is what they have to say about the vibration sensors intended to be used together with their circuit:


  The mechanism for detecting vibration of a shaft does not form a part of this invention which is directed to the objects set forth hereinbefore. Vibration detection mechanism, for use with the apparatus herein may be commercially obtained, for example, from the Bentley Nevada Corporation, an organization that markets a vibration detection pickup.


You can see the patent linked above for a complete description of the circuit, but it's obvious the negative supply is required for the correct biasing of its transistors (example).

One thing that supports this theory is the fact that well known early books on transistors, like Principles of Transistor Circuits by S.W. Amos, originally published in 1959 but re-edited and reprinted with the same content well into the 90s, used PNP transistors all throughout the book (see transistor symbol on its cover), contrary to most books published from the 70s on.

I understand that my answer might be just a narrative and there might have been many more factors at play. I do agree it seems strange to see the negative supply for eddy current sensors carried on all these years, something that did not happen for other similar devices; maybe I'm missing something...