Can I say that the two-body problem is also a vacuum solution for the special case of two orbiting black holes? A two body system is one where two bodies orbit each other. In the case of two orbiting black holes, since an individual black hole is described by a vacuum solution, can I say that the two-body problem is also a vacuum solution for the special case of two orbiting black holes?
 A: Yes and no.
A vacuum solution could be stationary, could be static, could be neither. For the orbiting black holes you end up with gravitational waves and gravitational radiation.
You either have some going out, and the bodies in spiraling or you have some coming in and driving the system or both. And the point is that you have to specify the state of the metric (and whether it has no radiation, incoming radiation, outgoing radiation, or orbiting radiation)  very far from both bodies.
So the radiation is a third object that is involved in the dynamics. Now since that is involved even in the zero body case you might argue it doesn't count. Its just confusing since people tend to eliminate it for the one body case 
So yes its just a vacuum solution to have two black holes. Yes they will orbit. And depending in the radiation the ride or give off they might in spiral at a certain rate ... or not.

So then it is a vacuum solution but the form of the metric depends on the radiation that is given off by the two bodies?

Radiation given. Radiation received. It all matters. And there is no obvious symmetry to help pick what radiation is out there.
The way matter curves spacetime is that you have a spacetime that is close to a vacuum Schwarzschild with mass $M$ outside one star and is a different a vacuum Schwarzschild with mass $m<M$ inside and there is energy of $(M-m)c^2$ on the surface where the two solutions are sewn together. That is what sources do, they allow different types of vacuum solutions to be sewn together. Then as the surface where the energy is contracts the vacuum Schwarzschild solution of the outside type fills in the region outside the energy where previous you had a different Schwarzschild solution.
So you orbit a black hole because you see the natural vacuum curvature that was left behind by the collapsing matter as it passed through that region.
So the exterior spacetime outside a black hole is the evolution of the curvature left behind by the star back when it passed through that region. All of space used to be full of energy back in the day. As it collapsed to form stars and black holes the in falling energy left the regions outside it more strongly curved than it used to be. The Schwarzschild solution of mass $M$ is very lightly curved far out where the gas forming the star started out but as the gas contracted that vacuum solution (the one with mass parameter $M$) covered a larger region a region that extended farther in. And that solution is more curved farther in. And it is more curved there than the mass $m$ solution is.
So all the curvature of the black hole was caused by in falling matter connecting that outside vacuum solution to different vacuum solutions. It was determined by the past behaviour of the in falling matter.
So a single Schwarzschild solution you can pick a particularly simple case where the vacuum has not radiation and is in fact completely static and stationary.
So the black holes are reacting to the previously curved spacetime determined by the past. And there is no obvious form for the curvature or the metric between the black holes when you have two black holes. Each should be reacting to the curvature of the other, but what exactly should be the metric between them? If would be like having a perfectly fine description of how people pay each other money based on what is in their many bank accounts but where exactly is their money initially distributed.
So what exactly is the initial metric between the two black holes. It could have outgoing radiation that hasn't escaped yet and is currently in between them. It could have radiation that is incoming. The far away region could have incoming radiation. It could have outgoing radiation. And it is a nonlinear theory the radiation doesn't just pass through each other.
Now in many situations you can argue that all the radiation so far is very very small and then you can evolve them and hope the metric between them becomes realistic before the radiation gets strong. But you definitely have to be able to model the radiation or else your results will not be accurate.
