Conclusion is alarmist. Web pages can't receive accelerometer or gyro events when they are in the background on iOS or Android. Applications can't receive those events in the background either on iOS (possibly also Android?)
Would it not be possible to present a sensitive page in a full-screen iframe, and then capture an approximation of what the user entered into that iframe by picking up accelerometer events from the outer page? Not a perfect hack by any stretch, since the URL would be wrong, but potentially problematic.
an Android service can access accelerometer data. not sure about background websites though.