What is involved in converting an AC electric motor, like the kind found in a vacuum cleaner, to run on DC power?
Usually the motors used in vacuum cleaners are universal, can work with both currents VDC and VAC.
And series wound universal motors will run on less than their design voltage (but slower) Unlike induction motors that tend to have excessive current draw if running too slow, series universal and permanent magnet DC motors handle speed control by voltage reduction pretty well.
Some AC motor designs will only operate on AC; others will operate on DC also. I don't know what your application is, but you might also consider converting your DC power to AC by running it through an electronic device called an inverter. This approach also allows you to step the voltage up with a transformer so the motor operates at its rated voltage. Converting to AC may also involve a few gotchas of its own. Induction motors, for example, are sensitive to both the frequency and the waveform of their input power and may overheat if fed with the wrong waveform even if it is at the correct voltage and frequency.