In my experience, they do.
Among nVidia GPUs, at least before the GeForce FX 5xxx (supporting OpenGL 2.0) support FBO, and I suspect that even older cards do.
Among the ATI GPUs old enough to support OpenGL 2.0, I saw GPUs like the HD 2400 and X1300, and they all support FBOs.
Among Intel GPUs, I think that basically the HD Graphics families have support for OpenGL 2.0, and all the HD GPUs I've seen support FBO. I also saw some other GPUs with support for 2.0 and FBO, including some versions of 965, and something called "Eaglelake". I'm not sure why only 965s support OpenGL 2.0. This may be a problem with the driver.
I, on the other hand, have not yet found 2.0 compatible GPUs that do not support FBOs.
I hope this purely empirical answer helps somewhat.
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