Electromagnetic fields carry/store energy, and in that capacity they like all other form of energy and mass, can distort spacetime. The problem is that in a typical laboratory environment, the distortion is undetectable. For example, say that you have light in a 100 centimeter, spherical cavity such that the steady-state electromagnetic energy in the cavity is 9 trillion joules. This is a horrendous amount of energy that would quickly evaporate the container. But never mind that, this is just an example, not an engineering prototype!
How much mass does 9 trillion joules correspond to? Well, from E = mc^2 9 trillion joules =9 x 10^19 ergs, and if you divide this by the speed of light squared, you find that all of this electromagnetic energy only amounts to 9 x 10^19/9 x 10^20 = 0.1 grams! By the way, this much energy is about equal to a small atomic bomb. Don't try this experiment at home.
Well, how much spacetime curvature will 0.1 grams of electromagnetic energy give you? Exactly the same as 0.1 grams of anything else! Do you feel perturbed? After all, you weigh 50 - 100 kilograms! Anyway, the corresponding spacetime curvature for 0.1 grams is equal to its equivalent Schwarschild radius ( black hole radius) which is about 10^-33 centimeters of distortion for every 0.00001 grams. This means 0.1 grams produces 10^-33 x 10000 = 10^29 centimeters of curvature. This is 1000 trillion times smaller than the nucleus of an atom.