Ignoring any damping forces acting on the column, when it is vibrating at its natural frequency, the total energy (kinetic energy + internal strain energy) is constant.
As you increase the compressive force and the natural frequency gets lower, you need less energy to make it vibrate with a given amplitude, because the velocity is smaller and therefore the KE is smaller.
When the frequency reaches zero, in theory you don't have to supply extra energy to start it "vibrating" with "infinite" amplitude (but infinitely slowly, of course!).
In practice, there is always some destabilizing force (caused by geometrical tolerances, off-center loads, etc) that can do work (force x distance) to start the "vibration".
Note, the above only applies to starting the "vibration" (or buckling). Once it has started, the behaviour is nonlinear and the motion is not "infinitely slow", as is obvious in real life when things buckle!