If gravitational waves come from two massive objects orbiting each other, then does the Earth-Moon system create gravitational waves? What frequency would they have?

All binary systems — massive bodies orbiting around each other, irrespective of what the bodies are — generate gravitational waves. Roughly speaking, the strength of the gravitational waves depends on the mass of the bodies, the distance from the observer to the binary, and the frequency — or period — of the waves, which is directly related to the orbital period (for a circular orbit, the gravitational wave period is half the orbital period).

If you think about the Earth-Moon system, the first consideration is that the mass is about 20,000,000 times lighter than the black holes that LIGO observed. Without changing anything else, that translates into gravitational waves roughly a trillion times weaker than those LIGO observed.

More importantly, the Moon’s orbital period is 27 days; the period of the gravitational waves generated by its orbital motion around the Earth is 13.5 days. LIGO is only sensitive to gravitational waves with periods between about 0.02 seconds and 0.001 seconds. The gravitational waves from the Earth-Moon system would be well outside the range of periods LIGO can detect.