Native bird laws tend to cover the entire US due to the migratory bird act. Most birds that do any traveling are protected across the whole country which makes it hard to own any of the native birds even if they aren't native to your state.
The greater roadrunner is the only species in the U.S. And they really do run down the roads all day, at least around here, lol. Neither type migrates. I'm guessing that the lesser roadrunner would probably not be a welcome guest in the U.S.