I saw a piano with this problem once that had been repaired in
the past.  The fixer used the old trick of putting on a larger pulley
on the bellows side, thus making in turn faster and suck harder,
overcoming the leaks in the system.  The player was an Ampico and he
probably did not want to remove the gazillion tacks in the bellows.
Anyway, by properly rebuilding the bellows and putting the properly
sized pulley on, the RPMs dropped and the shaking stopped.

Another possibility is that the legs themselves are not as solid as
they used to be.  Each glue joint needs to be strong.  Each leg must
be firmly attached to the piano.

Are the legs on the small and thin side, compared to other piano
models?  Perhaps they cannot absorb the vibration anyway.

Jeff Davis - in sunny Seattle

 [ The piano on its legs resonates like a tuning fork.  If the legs
 [ become loose the natural resonant frequency of the piano structure
 [ is lowered, to where it can be the same as the pump frequency.
 [ That's when the shimmy dance gets wild!  -- Robbie