Has anyone directly observed the equivalence of inertial and gravitational mass for the electromagnetic force? Before someone tells me to drop a bowling ball and feather off the Leaning Tower of Pisa in a vacuum, let me point out that, in general relativity, you can't deduce anything about either mass of a free-falling test particle, because it has zero true acceleration anyway.
So what we're left with is that gravitational mass can only be observed by the extent to which the object curves spacetime around it, i.e. by the motion of test particles in its vicinity.  Whereas inertial mass would have to be measured by applying an electromagnetic field, and measuring the true acceleration away from a geodesic.
But that means we can really only measure the gravitational mass of astro-sized objects.  But aren't all these objects pretty much neutral?  So it would be hard to measure their acceleration under an EM force ... My guess is the best candidate would be star systems with strong magnetic fields, but that doesn't sound like a very precise test.
And even if they turned out to converge within experimental error for stars, would that really prove anything about much smaller objects?  I mean, I'd buy the statement that they both increase monotonically as you add material to a given object, but that doesn't imply equality across the board.
Or if we can't compare them directly for a single object, what are the important theoretical reasons why we should believe they are equal?
 A: Yes. You are looking for tests of the weak equivalence principle. This has been experimentally confirmed down to a precision of $10^{-15}$. The specific experiment which achieved this level of precision was the so-called MICROSCOPE experiment
https://en.m.wikipedia.org/wiki/MICROSCOPE
This experiment used very sensitive accelerometers to measure the acceleration of two different test masses of different composition. The masses were moved using electrostatic repulsion. So this appears to fully satisfy your criteria.
I would also mention that prior to the MICROSCOPE experiment the previously most precise experiments were based on torsion balances. The forces generated by torsion are, at their core, electromagnetic also. So I would also have counted those experiments had you asked this question prior to the MICROSCOPE experiment.
A: As Dale points out there are tests of the weak equivalence principle that check that the acceleration of bodies of different composition, towards a large mass, are the same.
That checks that the 'passive gravitational mass' is equivalent to inertial mass.  The experiments show that the equivalence principle is true to a high degree of accuracy.
It means that one aspect of Einstein's General Relativity seems to be valid.  The motion of the masses can be modelled by a 'bending of space-time' by the large mass.
If you mean, in the question - Is there a way to check that the 'active gravitational mass' is equivalent to the inertial mass? Then it can't easily be accurately checked.
(the terms are explained here in the section weak equivalence principle)
It would mean checking that the amount that a mass bent space-time is proportional to it's inertial mass.  As you say we can't know the inertial mass from the passive gravitational mass, so another means (electrostatic) would be needed to determine it .
That seems to be limit the objects mass. Then it seems that, even if the gravitational attraction from that mass could be measured, the accuracy would be limited by our knowledge of $G$, unfortunately $G$ is only known to about $0.6$%.
It could well be that this type of experiment has never been carried out, there are likely to be large uncertainties. It would be interesting to know to what extent the active gravitational mass has been found to be equivalent to inertial mass.
Perhaps an expert in General Relativity will comment whether the equivalence follows once GR is accepted, although it would be a theoretical justification and not an answer to your 'Has anyone directly observed...' question.
