What carries electric field through space? A stationary charge "creates" a constant (but not uniform) electric field around it, and a moving charge "creates" a variable electric field around it. What "carries" the information about the existence of a stationary or a moving charge through space?
In particular, are photons necessarily involved in the process?
 A: Macroscopically we speak of electric and magnetic fields which follow the classical theory of Maxwell's equations.

A stationary charge "creates" a constant (but not uniform) electric field around it, and a moving charge "creates" a variable electric field around it. What "carries" the information about the existence of a stationary or a moving charge through space? 

In the classical theory there is no necessity for "carrying" stationary electric and magnetic fields, the equations attribute fields to charges and magnetic moments which need no "carrying". The fields describe the behavior of charges and dipoles . For variable fields, classical theory's mathematics accepts that they propagate as electromagnetic waves in vacuum with velocity c, with no need of a medium. The theory fits the data perfectly.

In particular, are photons necessarily involved in the process?

Classical electromagnetism is an emergent theory from the underlying quantum electrodynamics.. There the photon is an elementary particle of zero mass and spin 1, traveling with velocity c  and is the carrier of electromagnetic interactions, either in virtual form or real. An enormous number of photons build up the classcial electromagnetic wave, as demonstrated here.
So yes, photons are necessarily involved as the carrier of the EM interactions.
A: This question illustrates a crucial difference between fundamental fields and everyday ones.
Everyday fields are made by averaging over many particles. For example, consider a sound wave in air. It's nothing more than a ripple in the 'displacement field': it says that at one point, the air is stretched, while at some other point, it gets be squeezed. 
But if you zoom all the way in, to the level of individual air molecules, you'll find that nothing is getting stretched or squished at all! In fact, if you look at a single air molecule at some moment, you won't have any idea whether it's in a sound wave or not. We only see sound waves when we average over the positions over many air molecules; the individual molecules make up the field.
Fundamental fields, like the electromagnetic field, are not the same way. Such fields are not made of anything; we do not construct them out of smaller pieces! In particular, the electromagnetic field is not "made of" photons like air is made of air molecules. This misconception suggests that if you got rid of all the photons in a region, the electromagnetic field would cease to exist there, which is totally false. 
A better analogy is that photons are like sound waves. You can quiet a room, but the air will still be there; similarly, the electromagnetic field as an entity still exists when there are no photons.
So, to answer your question: the electromagnetic field carries electromagnetic waves. The waves are made of photons; they are not being carried by photons.
A: I would say photons as they are mediators of electromagnetic interaction. But I wouldn't say something is actually carrying information about the moving or stationary charge, it's just the field itself, as it is changing at the origin and is propogating at c. So the change of field at a certain point away from source is not immediate, there is a delay. And only after that time you find out that the charge is moving.
A: Currents and magnetic fields are what creates changes in electric fields. Without them, electric fields stay the same.
And it is electric fields that make magnetic fields change, without them, magnetic fields stay the same.
In particular here are the explicit equations for the electromagnetic field and $$\frac{\partial \vec E}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec J+\frac{1}{\mu_0}\vec \nabla\times \vec B\right)$$ tells the electric part of the field how to change and 
$$\frac{\partial \vec B}{\partial t}=-\vec \nabla\times \vec E$$ tells the magnetic part of the field how to change.
So if you had a charge that had always been at rest at the origin the magnetic field might initially be $\vec 0$ and the electric field might initially be $\frac{q\hat r}{4\pi \epsilon_0|\vec r|^2}$ and then if you moved the charge it would have a current which would make the electric field change and the new electric field could make the magnetic field change. The new magnetic field could make the electric field change and so on, like a ripple effect that expands at the speed of light. Each field changing in time based on the spatial variation of the field values around it.
