Energy and mass are equivalent. If the electric field stores energy, then it stores mass. Weight is proportional to mass, other things being equal, so the electric field will contribute to the weight of the capacitor.
Likewise a wound up mechanical watch has a greater weight than the same one unwound, other things (e.g. temperature) being equal.
It's really very simple, as explained in "Relativity Visualized" by Lewis Carroll Epstein.
N.B. It seems that professional relativists have been using a new convention regarding the terms "mass" and "energy". The physical laws are the same, so Epstein was not wrong, but it seems that his use of "mass" and "energy" (which matches Albert Einstein's use which is also out of date, it seems) is now frowned on as dated by professional relativists, or at least the majority of English speaking ones.
I will let my answer above stand, because it's not wrong, but allegedly uses "mass" and "energy" in a dated way. However I will add another version that uses terms that cannot be faulted by anyone.
In the old convention (Einstein's) mass and energy were the same thing. But in the new convention used by professional relativists since about 1970, "mass" means "rest mass" and "energy" means "total energy including rest mass".
If the electric field stores energy, or if there is some extra energy in or attached to the capacitor or its field or charge in any way, that extra energy means extra weight. The weight is given my W = mg and in this case the part called "m" is replaced by "E" for "energy", where E is the total energy including all forms of energy including KE and mechanical energy and including the rest mass. So we have W = Eg where E is the total energy in kilograms.
Likewise a wound up mechanical watch has a greater weight than the same one unwound, other things (e.g. temperature) being equal.
A simple and crystal clear explanation of this can be found it Lewis Epstein's wonderful book, "Relativity Visualized", but be warned that he uses the terms "mass" and "energy" in the way relativists including Einstein used them up until about 1970, it seems. It's not a problem, an in fact it might even be a strength as the new conventions are highly confusing, to me for one. One reason it is so confusing is that instead of using terms of art that are clear to all physicists and intelligent nonphysicists, such as "rest mass" (professional relativists call it just, "mass") and "total energy including rest mass" (they just say, "energy") they use in effect a private language, AKA "jargon" that not the majority of fellow physics graduates understand. In fact, the majority of them misunderstand it.
HardlyCurious raised a great point in his comment, which is that it is far from clear how the weight of the energy of the electric field pushes down on the capacitor. I don't know how, but I've read that it's figurative to say the energy of a capacitor is in the field.