Charles Torre, professor of physics at Utah State University and a researcher in gravitational and mathematical physics, provides this answer:
Image: History of Mathematics
ALBERT EINSTEIN'S theories explain how magnetic and electromagnetic fields induce curvature in space-time.

Electric charges and magnets do indeed "distort space," but this happens on a couple of levels.

First, a word of background. According to the current best theory of gravitation, which is contained in Albert Einstein's famous general theory of relativity, a gravitational field represents a curvature of space-time, rather than a distortion of it. Anything that carries energy, momentum and stresses is a source of a gravitational field, that is, a curvature of space-time.

Electric charges and magnets are manifestations of certain types of matter, most particularly electrons. Since matter carries energy (via Einstein's famous relation that energy is mass times the speed of light squared), such objects will have a gravitational field and so they will distort space-time. So one way in which a charge or a magnet will distort space-time is by virtue of its matter. That answer may not sound too impressive, but there is more. . . .

You see, electromagnetic fields themselves carry energy (and momentum and stresses). The energy density carried by an electromagnetic field can be computed by adding the square of the electric field intensity to the square of the magnetic field intensity. As another example, a beam of light (produced from, say, a laser) consists of an electromagnetic field, and it will exert a force on charged particles. Thus the electromagnetic field carries momentum. Because an electromagnetic field contains energy, momentum, and so on, it will produce a gravitational field of its own. This gravitational field is in addition to that produced by the matter of the charge or magnet.

A simple example of the gravitational (or space-time curvature) effect of electric charges arises in the "Reissner-Nordstrom" solution to Einstein's gravitational field equations. This solution describes the gravitational field in the exterior of a spherical body with non-zero net electric charge. (The solution describing the special case in which the net electric charge is zero is the famous "Schwarzschild solution" to the gravitational field equations.) From the Reissner-Nordstrom solution, it is clear that the motion of test particles in the gravitational field of the spherically symmetric body depends on whether or not the body carries a charge. Just as the Schwarzschild solution can be extended to describe the famous phenomenon of a "black hole," the Reissner-Nordstrom solution can be extended to describe a "charged black hole." For an electrically charged black hole, the gravitational field of the hole includes a contribution due to the presence of an electric field.

I do not know (and I doubt) whether this aspect of gravitational theory (that electromagnetic fields produce gravitational fields) has been directly tested by experiment. The difficulty is that the gravitational field produced by a typical electromagnetic field you can produce in a laboratory is predicted to be very, very weak. A better place to look for gravitational effects due to electromagnetic fields would be in astrophysical objects carrying a significant net electric charge. Unfortunately, to my knowledge, such objects are expected to be hard to come by. So while the answer to the question is definitely "yes" according to theory, the experimental status of this effect appears to be somewhat open.

If you want to read more about some of these ideas, you might try Space, Time, and Gravity, by Robert Wald (University of Chicago Press, 1992). This book is aimed at non-specialists. For a more detailed mathematical treatment, you can consult any text on the general theory of relativity.