Does electric charge vary between observers?

Charge is Lorentz invariant, meaning it is the same in all frames of reference. This means that four current is a four vector. This is because, for example, the time-like component is charge density, $\rho =\frac{dq}{dV}$. Because length only contracts in the direction of relative motion, volume only decreases by a Lorentz factor, the same as length. If charge is a scalar the charge density transforms as a component of a four vector. This gives Maxwell's equations their relativistic form

I can't speak with certainty but I imagine it is invariant under diffeomorphisms in general relativity too.

The core of the issue is the determination of inertial frames, where an inertial frame is defined as being a frame in which the laws of mechanics are the same. But, which laws? If it is Newtonian Mechanics then the Galilean transformation will shift from one frame in which Newtonian Mechanics works to another.

But, when electromagnetic theory is considered the Galilean transformation is wrong, it does not preserve Maxwell's equations.

The Lorenz transformation is one that does.

But, there are many other options.

The choice between them is the matter of a choice of a scalar gauge field. The Lorenz transformation is the one that also preserves the charge density.

To put a bit of context there, while a moving observer sees the equations of electromagnetic theory to be unchanged, they nevertheless see different electric and magnetic fields. In the general case of transformations that preserve the equations, the charge density - treated simply as a field of real numbers - also changes.

The Lorentz transformation is the gauge choice that preserves the speed of light. And it also preserves the charge density. This is not a coincidence, as there is, in the general case, a relation between the charge gauge (interpreted here as the charge on the electron, for example) and the speed of light.

Accelerating frames are more of a problem. You cannot preserve the electromagnetic equations in an accelerating frame. In physical terms, it is not inertial - so we should not expect the equations to be the same. This leads to interesting conundrums, paradoxes, and controversies.

If an accelerating electron radiates, does it radiate in its own frame? Since gravity is equivalent to acceleration, should not an electron held stationary on the surface of the Earth radiate - due to that acceleration?