I'm thinking that Doppler shift only applies to a source that emits electromagnetic waves, like stars and galaxies. How about a black hole? Since we can't detect Hawking radiation, do we have a reliable way to find out (assuming it doesn't belong to a binary stars system)?
If the black hole is accreting material, a disc will form around it (perhaps more like a torus; the centre must be empty because there is a black hole in the way). Because of the immense forces and high velocities involved, the disc will be very hot, with a gradient to cooler temperatures further out. The inner edge is usually hot enough to emit X-rays, and there is usually some iron in the gas - this leads to the formation of a nice feature in the X-ray spectrum called the iron K-line, which can be used to measure a doppler shift. There will in general be two components to the shift, one from the approaching/receding side of the disc which, if the source is unresolved, will broaden and/or split the line, and one from the net approaching/receding velocity of the black hole, which is what you asked about.
The caveat to the above is that for a stellar mass black hole to be accreting enough material, it must have material donated from a companion star and thus be in a binary (these systems are members of the well-known X-ray binary class of objects). Supermassive black holes, however, often have an accretion disc fed by the ISM of their host galaxy. One can usually infer that the velocity of the black hole will be the same as the velocity of the galaxy in this case, and the velocity of the galaxy is usually much easier to measure, but there are some supermassive black holes with velocities offset from their host galaxies, which are interesting systems.
One last point to make a pedantic correction to a statement in your question. You say Doppler shift only applies to sources emitting electromagnetic waves; more accurately it applies to a source emitting any kind of wave, including acoustic (black holes don't emit any detectable acoustic waves) and gravitational waves (which a merging black hole + black hole binary can emit strongly enough to be detected). The trouble is that gravitational wave spectra don't yet have any useful known spectral features that might allow us to measure a Doppler shift (though interestingly the luminosity distance of a merging black hole event can be measured, which can be converted to a cosmological redshift via Hubble's law). Anyway I'm starting to get rather off topic, I'd better stop here.