Do black holes have energy?

An isolated black hole is a vacuum solution of general relativity, so in a very direct sense it does not contain any energy anywhere in spacetime. But perhaps somewhat counter-intuitively, that does not imply that such a black hole has no energy.

Defining the total amount of energy is usually very problematic in general relativity, but in some special cases it is possible. In particular, the usual black hole solutions are all asymptotically flat, i.e., spacetime is just the usual flat Minkowski when far away from the black hole.

Here (or in general when we have a prescribed asymptotic form of the spacetime), we can calculate the total energy-momentum, by essentially measuring the gravitational field of the black hole at infinity. The energy just be one component of energy-momentum.

There are actually two relevant different kinds of 'infinity' here: spatial infinity and null (lightlike) infinity, depending on whether we are 'far away' from the black hole in a spacelike or lightlike direction. There's also timelike infinity, but that just corresponds to waiting an arbitrarily long time, so it's not relevant here. The two different infinities beget different definitions of energy-momentum, giving the the ADM energy and the Bondi energy, respectively. In a vacuum, the intuitive difference between the two is that Bondi energy excludes gravitational waves.

So the short answer is 'yes', with the caveat that in a more complicated situation, where we can't attribute everything to the black hole itself, the answer to how much energy is due to the black hole may be ambiguous or ill-defined.

Note that the ADM and Bondi energy-momenta also define their corresponding measures of mass, as the norm of those energy-momenta ($m^2 = E^2-p^2$), but for a black hole we can also define mass more operationally in terms of orbits around the black hole. There are also other alternatives for addressing mass specifically.