Relativisitc mass is just another name for energy, so a light wave has some. However its wrong to ascribe gravity as being a force due to relativistic mass (though if you're lucky you'll be within a factor of 2 of the correct answer. For instance, you'll get only half the expected deflection of light by using a quasi-Newtonian approach.)
"Relativistic mass" is an SR concept that is not particularly useful in GR, and pretty much of a dead end. The good news is it can be defined, the bad news is it's not particularly good for anything, including computing gravity, and the things that it can do can be done just as well if not better by other means.
So if you're asking about the relativistic mass of light in order to compute the gravitational effect of light, you're asking the wrong question.
Many modern SR treatments of SR don't even bother with relativistic mass anymore, though you'll find the occasionall odd enthusiast of the topic, and an occasional textbook reference (mostly in older textbooks).
"Relativistic mass" seems very appealing to the mass media and lay audience more than it's use in the professional literature for some reason.
Good relativistic treatments of mass by General Relativity are a fairly advanced subject in general relativity. In the current state of the art, there is not one, but several competing definitions of mass, each of which can be applied to their own class of problem.
The Komar mass, which applies to static space times, is one of the simplest. A few other examples are ADM and Bondi masses, and there are others, though these are probalby the 3 most common.