The question is whether the answers are true where it is really important, in General Relativity, and not just Newtonian physics.
The answers provided so far are correct. Electric charge does not couple directly to the gravitational field, but rather to the electromagnetic field. The equivalence principle in essence says that the source of the gravitational field, the gravitational mass, is proportional to the inertial mass, and thus one cannot distinguish uniform gravitational fields from accelerated frames. The same is not true or electric charge, it is not proportional to inertial mass (it is an independent property of a particle instead), and thus acceleration will actually depend on q/m, the charge to mass ratio.
This explains the a argument for classical Newtonian physics, where the equation F = ma comes from and is true. But there is no such equation in General Relativity (GR). So the question is, do we know that the same is true in GR? We know it should be, that GR was constructed to obey the equivalence principle for gravitational fields. How were other forces (or fields) accounted for, and is motion again dependent on q/m?
The answer is yes, geodesic motion in GR with an electromagnetic field which contributes to the motion, for a charged particle, just depends on q/m.
The way that comes about is how Maxwell's equations, and the equations of motion, were included in the so called Einstein Maxwell equations. The Maxwell's equations were included in the Einstein equations as part of the right hand side, the stress energy tensor, and the covariant Maxwell's equations.
The Einstein Maxwell equations, and the covariant Maxwell's equations ('simply' replace commas by semicolons, i.e., make derivatives covariant) are in
The right hand side is how the electromagnetic field contributes to the gravitational field. The spacetime is called an electrovac spacetime
The next paper derives and describes, in detail, the geodesics in the resulting electrovac spacetime produced by charged black holes (they did it for static, rather than stationary spacetimes, but it's been done for stationary Kerr Newman spacetimes also).
It describes test particle motion in Einstein Maxwell spacetimes. Depends on particle charge (and theoretically magnetic charge, if it existed). Mass gets normalized out, i.e. It's the ratios of q/m and theoretically $q_m$/m, the magnetic charge to mass ratio, if magnetic charge exists, as well as the metric, that determines the motion of the particles.
So, yes, it is built into 4D GR.
It is interesting that in 5D, as is well known, one can actually get an equivalence principle. In 4D it is instead q/m, as in classical physics.